That applies even when evidence not in the state court record shows a constitutional violation, the ruling said.
Four justices joined in the prevailing opinion, several joined in parts of it and at least one, Justice Sonia Sotomayor, dissented to the whole thing.
The ruling came in the case of Scott Lynn Pinholster, who was convicted of murdering two men in a Los Angeles robbery that netted $23 and a tiny bit of marijuana.
During the penalty phase, the jury heard eight eight witnesses who testified about Pinholster's violent past behavior. His attorney objected, saying prosecutors had not given him notice of the aggravating evidence as required by California law.
Though the defense consulted a psychiatrist, he was not put on the stand. Only Pinholster's mother testified for him during the penalty phase.
The jury recommended the death penalty, and the state judge sentenced him to death.
When two petitions to the California Supreme Court to overturn the death penalty failed, despite additional evidence allegedly showing Pinholster had a personality disorder, his lawyers asked for constitutional review in federal court.
A U.S. judge and appeals court granted Pinholster review in light of the new evidence.
The U.S. Supreme Court reversed, saying federal court review generally is limited under U.S. law "to the record that was before the state court that adjudicated the claim on the merits."
Under the 1995 Anti-terrorism and Effective Death Penalty Act -- which limits appeals, especially death row appeals that had been going on for decades -- federal review "shall not be granted ... unless the adjudication ... resulted in a decision that was contrary to, or involved an unreasonable application of, clearly established federal law," or "resulted in a decision that was based on an unreasonable determination of the facts in light of the evidence presented in the state court proceeding."
Justice Clarence Thomas wrote the opinion, which was joined by three fellow conservatives. Fellow conservative Justice Samuel Alito joined for most of it, and concurred in the judgment; liberal Justices Stephen Breyer, Ruth Bader Ginsburg and Elena Kagan joined parts.
Sotomayor, joined in part by Ginsburg and Kagan, filed a dissenting opinion.
"Under the (court majority's) novel interpretation of (the U.S. law on appeals) ... federal courts must turn a blind eye to new evidence in deciding whether a petitioner has satisfied (the federal law's) threshold obstacle to federal habeas (constitutional) relief -- even when it is clear that the petitioner would be entitled to relief in light of that evidence."
Gal Gadot cast as Wonder Woman for 'Batman vs. Superman'
Exploding whale video goes viral on Internet