MINER, Circuit Judge:

Following a jury trial, judgment was entered in the United States District Court for the Southern District of New York (Sprizzo, J.) in favor of plaintiff James Staffer for $480,650.38 plus costs and interest against Bouchard Transportation, Inc. and Frederick E. Bouchard, Inc. (collectively "Bouchard"), his employers, for injuries he suffered while working as a deckhand. Following a second jury trial in the same court, judgment was entered against third-party defendant Joseph Suarez, M.D. in favor of Bouchard for contribution in the sum of $462,489.98 plus costs and interest as a result of the malpractice of Suarez in the treatment of Staffer. The judgment incorporated a finding of no liability on the part of third-party defendant Staten Island Hospital. After entry of a consolidated judgment, Staffer sued the hospital and Suarez in state court for medical malpractice. A motion in federal court by the hospital and Suarez to enjoin the state court proceedings on the ground of res judicata was denied. 697 F.Supp. 765.

The hospital and Suarez appeal the denial of their motion to enjoin. Suarez also appeals the award in favor of Bouchard, arguing that he should be entitled to set off from it any monies Staffer has received in maintenance and cure from Bouchard and any social security disability benefits that Staffer has received and will receive. Additionally, Bouchard appeals the denial of its motion for reimbursement from Suarez of attorney's fees incurred in prosecuting its third-party claim.

Because an injunction to prevent Staffer's malpractice claim against the hospital and Suarez in state court is unwarranted under the Anti-Injunction Act, 28 U.S.C. § 2283 (1982), we hold that the district court's refusal to enjoin the state proceedings does not require reversal. We perceive no basis in New York law to set off from Suarez' obligation to Bouchard any collateral source income Staffer may have received. Finally, we decline to award attorney's fees to Bouchard.

BACKGROUND

On July 3, 1985, James Staffer injured his right shoulder while releasing the towing machine brake aboard the tugboat FREDERICK E. BOUCHARD. At the time of the accident Staffer was employed as a deckhand by Bouchard. Staffer has not been able to work as a deckhand since the mishap. Pursuant to procedures established by a collective bargaining agreement, Bouchard paid Staffer a total of $68,403.35 in maintenance and cure. As of December 1986, Staffer had received an additional $8034 in social security disability benefits. Since then, he has received in such benefits $773 monthly; the monthly payments will continue for the rest of his life.

On July 7, 1986, Staffer initiated a personal injury action against Bouchard, pursuant to the Jones Act, 46 U.S.C.App. § 688 (1982), in the United States District Court for the Southern District of New York (Sprizzo, J.). By a third-party complaint filed on March 20, 1987, Bouchard impleaded the Staten Island Hospital and Dr. Joseph A. Suarez, see Fed.R.Civ.P. 14, alleging medical malpractice that aggravated Staffer's injuries. Staffer did not amend his complaint to assert a claim against the hospital or Suarez. Judge Sprizzo ordered a severance, and the two claims were tried by separate juries.

In Staffer's trial against Bouchard, the jury returned a verdict of $739,462, but found that Staffer had been 35% contributorily negligent. Accordingly, judgment of $480,650.38 plus costs and interest was entered in favor of Staffer against Bouchard on January 5, 1988. In Bouchard's third-party action against the hospital and Suarez, the jury found that Suarez had been negligent and that his negligence contributed 100% to Staffer's pain and suffering and 90% to his physical injury. Judgment of $462,489.98 plus costs and interest was entered in favor of Bouchard against Suarez on May 31, 1988. The claim against the hospital was dismissed and the hospital was awarded costs against Bouchard. Suarez' motion for an offset of his contribution pursuant to New York's collateral source statute, N.Y.Civ.Prac.L. R. § 4545(a) (c) (McKinney 1989) [hereafter CPLR], which generally forbids in certain malpractice and tort actions recovery of damages that plaintiff already has recovered or will recover from collateral sources, was denied. The court denied also Bouchard's motion for attorney's fees against Suarez. With this second judgment, Judge Sprizzo vacated the earlier judgment resolving the Staffer-Bouchard conflict and incorporated all verdicts and assessments into one final judgment.

Shortly after the verdict on Bouchard's third-party action, Staffer brought a malpractice action against the hospital and Suarez in New York State Supreme Court. The hospital and Suarez removed the action to the United States District Court for the Eastern District of New York (McLaughlin, J.), see 28 U.S.C. § 1441(a) (1982), on the ground of federal question jurisdiction, claiming they were pendent parties to the earlier action in the Southern District of New York. They then moved to transfer the action to the Southern District of New York, see 28 U.S.C. § 1404(a) (1982), to consolidate it with the earlier action, see Fed.R.Civ.P. 18, in which Judge Sprizzo had not yet entered his revised judgment. Judge McLaughlin, however, granted Staffer's cross-motion to remand the case to state court, holding that the removal to federal court had been improvident and without jurisdiction. See 28 U.S.C. § 1447(c) (1982) (since amended by the Judicial Improvements and Access to Justice Act of 1988, 28 U.S.C.A. § 1447(c) (West Supp. 1989)).

Thereafter, the hospital and Suarez moved in the Southern District of New York (Sprizzo, J.) to alter or amend the federal court judgment, see Fed.R.Civ.P. 59(e), to enjoin further action by Staffer in state court on the ground of res judicata. Judge Sprizzo denied the motion, holding that Staffer had been under no obligation to assert a direct claim against the hospital and Suarez in his initial federal action.

Staffer then moved in state court for summary judgment. He sought damages of $222,022.96, approximately what he had been unable to recover in his action against Bouchard because of contributory negligence. Staffer argued that his contributory negligence was not available to the defendants as a defense in the malpractice action and that Suarez' liability already had been determined in Bouchard's third-party action. The hospital and Suarez cross-moved in state court for (i) a stay until this appeal could be decided, (ii) summary judgment dismissing the complaint on the ground of res judicata because Staffer could have raised his malpractice claim in his original federal action but did not, or (iii) denial of the portion of Staffer's motion for summary judgment that sought to recover lost wages and other special damages subject to CPLR § 4545.

The state court, agreeing that the issue of malpractice had already been decided, granted Staffer's motion and entered an order in his favor, awarding him damages, less any amount determined by further discovery to be subject to the exclusion of CPLR § 4545. Similarly, the cross-motion to limit the amount of Staffer's judgment was granted to the extent that the damages award should be reduced by operation of section 4545. The state court declined to stay its proceedings but granted the cross-motion for summary judgment as to the hospital, dismissing the complaint against it with prejudice; Staffer consented to this dismissal. The state court denied the cross-motion for summary judgment as to Suarez. Later, on March 13, 1989, the state court entered judgment against Suarez and in favor of Staffer in the amount of $222,022.96 plus interest, apparently considering any section 4545 setoff inappropriate.

The hospital and Suarez appeal the district court's denial of their motion to enjoin the state action, and Suarez challenges the amount the district court required him to pay on the third-party malpractice claim, seeking a setoff for maintenance and cure and social security received or to be received by Staffer. Bouchard cross-appeals from the district court's judgment, requesting attorney's fees for its third-party claim against Suarez.

DISCUSSION

1. Mootness

As an initial matter, we find that the issue of whether the district court properly declined to issue the injunction has not been rendered moot by the judgment entered against Suarez in state court. The state court proceedings are not completed until the process of redressing wrongs is finalized. See Hill v. Martin, 296 U.S. 393, 403, 56 S.Ct. 278, 282, 80 L.Ed. 293 (1935). Here, according to the record, further proceedings relating to appeal and collection may yet be instituted. Because an injunction would halt those proceedings, our jurisdiction over this matter is intact. See Deposit Guar. Nat'l Bank v. Roper, 445 U.S. 326, 333, 100 S.Ct. 1166, 1171, 63 L.Ed. 2d 427 (1980) (payment and satisfaction of final, unappealable judgment renders controversy moot).

2. Anti-Injunction Act

A federal court generally is prohibited by the Anti-Injunction Act from enjoining proceedings in state court. 28 U.S.C. § 2283 (1982). This "forestall[s] the inevitable friction between the state and federal courts that ensues from the injunction of state judicial proceedings by a federal court." Vendo Co. v. Lektro-Vend Corp., 433 U.S. 623, 630, 97 S.Ct. 2881, 2887, 53 L.Ed.2d 1009 (1977) (plurality). By virtue of the Act's "relitigation exception," however, a federal court may halt state litigation that might undermine some of the res judicata or collateral estoppel effects of a federal judgment. Chick Kam Choo v. Exxon Corp., 486 U.S. 140, 108 S.Ct. 1684, 1689-90, 100 L.Ed.2d 127 (1988). But, for this exception to be properly exercised, it is an "essential prerequisite ... that the claims or issues which the federal injunction insulates from litigation in state proceedings actually have been decided by the federal court . . . . [T]his prerequisite is strict and narrow." Id., 108 S.Ct. at 1690.

The district court did not err when it refused to enjoin the state court from entertaining Staffer's claim against the hospital and Suarez; the relitigation exception, as founded on principles of res judicata, could not properly have been invoked. Res judicata, or claim preclusion, bars the revival of claims that already have been litigated. Federated Dep't Stores, Inc. v. Moitie, 452 U.S. 394, 398, 101 S.Ct. 2424, 2427, 69 L.Ed.2d 103 (1981). Certainly, Staffer's malpractice claim against the hospital and Suarez was not raised in or decided by the federal court. Specifically, the issue of damages owed to Staffer by the hospital and Suarez, an essential element of Staffer's overall claim against the hospital and Suarez, had not been litigated or resolved in federal court. Furthermore, Staffer was not a party to the malpractice action in federal court. See Montana v. United States, 440 U.S. 147, 154, 99 S.Ct. 970, 974, 59 L.Ed.2d 210 (1979) (res judicata cannot bind those not party to prior adjudication).

Res judicata can also operate to bar claims that were not originally asserted. Federated, 452 U.S. at 398, 101 S.Ct. at 2427 ("A final judgment on the merits of an action precludes the parties or their privies from relitigating issues that ... could have been raised in that action."); Browning Debenture Holders' Comm. v. DASA Corp., 605 F.2d 35, 39 (2d Cir. 1978). The hospital and Suarez argue that because Staffer could have asserted his malpractice claim against them in his initial action in federal court, as he was authorized to do by Fed.R.Civ.P. 14(a), he should be barred from doing so in state court by the final judgment in federal court, and Judge Sprizzo should have enjoined the state court proceedings. Staffer responds that Rule 14(a) is permissive only and he was not required to amend his original complaint and assert a direct claim in federal court against the hospital and Suarez. While Staffer's argument has some merit, we rest our decision on the Anti-Injunction Act, which bars the issuance of an injunction under the circumstances revealed here.

The language and facts of Chick Kam Choo make clear that the part of the relitigation exception to the Anti-Injunction Act that is based on concerns of res judicata is more narrowly tailored than the doctrine of res judicata. The relitigation exception does not protect the full res judicata effect of a federal court's judgment; rather, it protects only matters that actually have been decided by a federal court. See Delta Air Lines, Inc. v. McCoy Restaurants, Inc., 708 F.2d 582, 586 (11th Cir. 1983) (declining to enjoin state action where claim could have been, but was not, brought in previous federal action). Undoubtedly, res judicata can bar claims that might and should have been raised but were not, see Federated, 452 U.S. at 398, 101 S.Ct. at 2427, but, for Anti-Injunction Act purposes, only "relitigation" can be enjoined. Here, the proper forum for a complete investigation of the res judicata effects of the district court's judgment is the state court, "which [is] presumed competent to resolve" such matters, Chick Kam Choo, 108 S.Ct. at 1691; see also Delta Air Lines, 708 F.2d at 586 (assertion of res judicata should be directed to state forum). The district court properly declined to enjoin completely the state court proceedings.

The district court, however, had the discretionary authority to enjoin the state court proceedings for jeopardizing the collateral estoppel effect of its judgment. Generally, collateral estoppel, or issue preclusion, bars relitigation of issues that actually and fairly have been decided in a prior action. See 18 C. Wright, A. Miller E. Cooper, Federal Practice and Procedure § 4416, at 137-38 (1981). The issue of the malpractice of the hospital and Suarez was litigated in federal court with Bouchard's claim for contribution. Suarez and the hospital had the opportunity to be heard in defense of the allegation of malpractice and defended with full vigor. See Allen v. McCurry, 449 U.S. 90, 95, 101 S.Ct. 411, 415, 66 L.Ed.2d 308 (1980) ("collateral estoppel cannot apply when the party against whom the earlier decision is asserted did not have a `full and fair opportunity' to litigate"). The malpractice issue litigated in and decided by the federal court is identical to that raised by Staffer in state court. See Brown v. Felsen, 442 U.S. 127, 139 n. 10, 99 S.Ct. 2205, 2213 n. 10, 60 L.Ed.2d 767 (1979) ("collateral estoppel treats as final only those questions actually and necessarily decided in a prior suit"); Thistlethwaite v. City of New York, 497 F.2d 339, 341-42 (2d Cir.) (only issues identical to those already tried may be collaterally estopped), cert. denied, 419 U.S. 1093, 95 S.Ct. 686, 42 L.Ed.2d 686 (1974). See generally G. Shreve P. Raven-Hansen, Understanding Civil Procedure §§ 111-113, at 433-46 (1989).

To protect the collateral estoppel effect of its judgment, see Parklane Hosiery Co. v. Shore, 439 U.S. 322, 99 S.Ct. 645, 58 L.Ed.2d 552 (1979) (collateral estoppel can be used by plaintiff not party to first action), the district court could have enjoined the state court from allowing Staffer to proceed with litigation on the malpractice issue only, while refusing to grant the entire relief sought by the hospital and Suarez. However, that it did not is, at this point, of no consequence. The state court granted collateral estoppel effect to the district court's resolution of the malpractice issue — precisely what injunction under the relitigation exception of the Anti-Injunction Act was designed to achieve. "Challenges to the correctness of a state court's determination as to the conclusive effect of a federal judgment must be pursued by way of appeal through the state-court system," not by the "highly intrusive remedy of a federal-court injunction." Parsons Steel, Inc. v. First Alabama Bank, 474 U.S. 518, 525, 106 S.Ct. 768, 773, 88 L.Ed.2d 877 (1986).

Furthermore, injunction of state court proceedings, even where warranted, is discretionary. See Chick Kam Choo, 108 S.Ct. at 1692; Delta Air Lines, 708 F.2d at 587. Indeed, the Supreme Court has admonished lower courts to refrain from enjoining state court proceedings unless absolutely necessary. See Atlantic Coast Line R.R. v. Brotherhood of Locomotive Engineers, 398 U.S. 281, 297, 90 S.Ct. 1739, 1748, 26 L.Ed.2d 234 (1970). Accordingly, we decline to reverse the district court's denial of the motion to enjoin.

3. Damages

Suarez, citing New York law, contends that he is entitled to set off from the judgment entered in federal court against him and in favor of Bouchard the maintenance and cure payments and social security benefits that Staffer already has received and the social security benefits that Staffer will receive in the future. Suarez argues that without such a setoff Staffer will be allowed double recovery. We are not persuaded that setoff is appropriate in this case.

Maintenance and cure, a right under general maritime law, gives to a seaman, injured in the service of a ship, wages to the end of the voyage as well as subsistence, lodging and care to the point where the maximum cure attainable has been reached. See Calmar S.S. Corp. v. Taylor, 303 U.S. 525, 530-32, 58 S.Ct. 651, 654-55, 82 L.Ed. 993 (1938); The Osceola, 189 U.S. 158, 175, 23 S.Ct. 483, 487, 47 L.Ed. 760 (1903). A claim for reimbursement of maintenance and cure, even when it is the subject of a third-party action, "may be pursued as a matter of general maritime law." Black v. Red Star Towing Transp. Co., 860 F.2d 30, 34 (2d Cir. 1988) (in banc).

While federal law can accord third-party plaintiffs the right to contribution, id., the actual method, rules and application of contribution in this case are a matter of state law, see 28 U.S.C. § 1652 (1982). By federal law, Bouchard is permitted contribution. However, state law, namely CPLR § 4545, governs the extent of any setoff sought by Suarez against the award to Bouchard. CPLR § 4545(a) reads, in part:

In any action for medical . . . malpractice where the plaintiff seeks to recover for the cost of medical care, ... custodial care or rehabilitation services, loss of earnings or other economic loss, evidence shall be admissible for consideration by the court to establish that any such past or future cost or expense was or will, with reasonable certainty, be replaced or indemnified, in whole or in part, from any collateral source . . . . If the court finds that any such cost or expense was or will, with reasonable certainty, be replaced or indemnified from any collateral source, it shall reduce the amount of the award by such finding . . . .

CPLR § 4545(c) employs nearly identical language, the only difference being that it applies to "any action brought to recover damages for personal injury, injury to property or wrongful death."

Under New York law, there is no basis for offsetting from any jury award Staffer's maintenance and cure or social security benefits. Bouchard is neither a plaintiff in a malpractice action within the meaning of CPLR § 4545(a) nor a plaintiff in a personal injury action within the meaning of CPLR § 4545(c). Bouchard is a third-party plaintiff seeking contribution from Suarez. While Bouchard's theory of contribution encompasses an assertion of malpractice and injury by Suarez, the nature of Bouchard's action is contribution. Cf. Bay Ridge Air Rights, Inc. v. New York, 44 N.Y.2d 49, 53, 375 N.E.2d 29, 30, 404 N.Y.S.2d 73, 74 (1978) (claim for contribution accrues not as of date of tort but when liability of party seeking contribution is satisfied by payment).

Furthermore, as to maintenance and cure, there is no possibility of double recovery by either Staffer or Bouchard. Staffer's jury award did not include maintenance and cure. He had been paid maintenance and cure by Bouchard and so did not seek it in his legal action. The special verdict for damages submitted by the jury in the trial between Staffer and Bouchard clearly indicates that the award is based solely on "future lost earnings" and "past, present, and future pain and suffering together with loss of life's pleasures." Nor has Bouchard received reimbursement from a collateral source for the payments of maintenance and cure it has made to Staffer. Absent any "reimbursement or indemnification from a collateral source," there is no reason to reduce Suarez' obligation to Bouchard for Staffer's maintenance and cure.

4. Attorney's Fees

Bouchard, on cross-appeal, seeks reversal of the district court's denial of its motion for attorney's fees. Bouchard argues that it is entitled to attorney's fees by way of indemnity, because Suarez was constructively its agent and his malpractice was a breach of implied warranty. We are not persuaded by Bouchard's theory of recovery; nor do we find any contractual basis for such an award, see Black, 860 F.2d at 34-35. Accordingly, we decline to award attorney's fees.

CONCLUSION

The judgment of the district court is affirmed.