The decision may have limited impact because the federal government and 47 states already have laws that allow convicts some access to genetic evidence. Testing has led to the exoneration of at least 232 people who had been found guilty of murder, rape and other violent crimes.
The court ruled 5-4, with its conservative justices in the majority, against an Alaska man who was convicted in a brutal attack on a prostitute 16 years ago.
"It's hard to know whether this ruling is going to speed up the national trend toward more DNA testing," writes CBS News chief legal analyst Andrew Cohen. "Even many prosecutors are pushing for it, or at least allowing it, in certain cases where it would be relevant. In the meantime, 13 innocent defendants have been exonerated this year alone."
William Osborne won a federal appeals court ruling granting him access to a blue condom that was used during the attack. Osborne argued that testing its contents would firmly establish his innocence or guilt.
Separately, in parole proceedings, Osborne has admitted his guilt in a bid for release from prison.
The high court reversed the appellate ruling. States already are dealing with the challenges and opportunities presented by advances in genetic testing, Chief Justice John Roberts said in his majority opinion.
"To suddenly constitutionalize this area would short-circuit what looks to be a prompt and considered legislative response," Roberts said.
But Justice John Paul Stevens said in dissent that a simple test would settle the matter. "The court today blesses the state's arbitrary denial of the evidence Osborne seeks," Stevens said.
In other Supreme Court rulings:
The court, in a 6-3 vote, threw out an appeals court ruling that would have allowed a retrial of F. Scott Yeager, a former executive at Enron's failed broadband venture, on charges for which a jury could not reach a verdict at his first trial.
Travelers had been named in lawsuits alleging that it tried to hide dangerous health effects of asbestos. The company argued that asbestos claims be paid out of a trust created by Johns Manville in the 1980s and approved by a federal bankruptcy judge.
Travelers settled with several groups of plaintiffs with the caveat that federal courts make clear the company would not have to face any new similar lawsuits.
The 2nd U.S Circuit Court of Appeals in New York overturned lower-court approval of the settlement, saying a bankruptcy judge lacks the authority to act so broadly.
The high court overturned that decision.