WASHINGTON — Two weeks ago, on the same day it heard arguments about the future of abortion rights in Texas, the Supreme Court turned down an appeal from a federal prisoner facing execution.

The move was in one sense routine, as the court has grown increasingly hostile to arguments made by death row inmates. This became apparent in the final months of the Trump administration, when, after a hiatus of 17 years, the federal government executed 13 inmates.

“Throughout this expedited spree of executions, this court has consistently rejected inmates’ credible claims for relief,” Justice Sonia Sotomayor wrote in a dissent at the time.

The court’s impatience was also evident last week at an argument over whether an inmate’s pastor could pray with and touch him in the death chamber. Several conservative justices expressed dismay at what they said was last-minute litigation gamesmanship in death penalty cases.

Still, the case the court turned down two weeks ago was exceptional, providing a telling glimpse of the state of capital punishment in the United States. The court rejected the inmate’s petition even though the prosecution agreed that his case deserved a fresh look.

In an 11-page dissent, Justice Sotomayor, joined by Justices Stephen G. Breyer and Elena Kagan, said the majority had crossed a new bridge.

“To my knowledge, the court has never before denied” such relief “in a capital case where both parties have requested it, let alone where a new development has cast the decision below into such doubt,” Justice Sotomayor wrote.

The case concerned Wesley P. Coonce Jr., who was serving a life sentence for kidnapping and carjacking when he helped murder another prisoner in the mental health ward of a federal prison. A murder committed by an inmate already serving a life sentence is a capital crime, and he was sentenced to death.

Lawyers for Mr. Coonce asked the justices to return his case to an appeals court for reconsideration of his argument that he could not be executed because he was intellectually disabled. There had been, the lawyers wrote, an important new development that could alter the appeals court’s analysis.

The prosecution supported the request.

“The government agrees” that a fresh look “would be appropriate in light of a significant intervening factual change that affects a central predicate of the court of appeals’ Eighth Amendment analysis,” wrote Elizabeth B. Prelogar, who has since been confirmed as the U.S. solicitor general.

The Supreme Court has ruled that the Eighth Amendment, which bars cruel and unusual punishment, forbids the execution of people who are intellectually disabled. The appeals court had ruled that Mr. Coonce did not qualify under the criteria set out in a 2002 Supreme Court decision, Atkins v. Virginia, which drew on clinical definitions of intellectual disability.

The Atkins decision defined intellectual disability to have three components: low IQ scores; a lack of fundamental social and practical skills; and the presence of both before the age of 18.

There was substantial evidence that Mr. Coonce satisfied the first two prongs of the test. But his intellectual deficits were apparently the result of a traumatic brain injury when he was 20.

Before the appeals court, Mr. Coonce’s lawyer had argued that the professional consensus on that third prong was evolving. A leading professional organization, the American Association on Intellectual and Developmental Disabilities, they wrote, was considering changing its definition.

That was not enough, a unanimous panel of the U.S. Court of Appeals for the Eighth Circuit ruled. The court said it would not consider “predictions that medical experts will agree with Coonce’s view in the future,” adding that “such evidence is not sufficient for us to divine any current Eighth Amendment limitation.”

After Mr. Coonce asked the Supreme Court to hear his case, the association revised its criteria, saying the relevant developmental period extended to the age of 22.

What had been a prediction had ripened into reality. But the Supreme Court declined to let the appeals court assess the significance of the development.

While the majority did not explain its thinking, a 2014 dissent from Justice Samuel A. Alito Jr., joined by Chief Justice John G. Roberts Jr. and Justice Clarence Thomas, provided a hint. Justice Alito wrote that the meaning of the Eighth Amendment should not be determined by “positions adopted by private professional organizations.”

The majority may also have thought that the Biden administration had its own tools to address Mr. Coonce’s case, notably by granting him clemency.

A few days before the court turned down Mr. Coonce’s appeal, it lifted a stay of execution in a case from Oklahoma, which has a history of botched executions, again over the dissents of Justices Breyer, Sotomayor and Kagan. A federal appeals court had imposed the stay so that it could examine an argument from two death row inmates who said that requiring them to specify an alternative method of execution in order to challenge the state’s chosen method was at odds with their faith.

One of the men, John Marion Grant, who had been convicted of murdering a prison cafeteria worker, was executed a few hours later, convulsing and vomiting as the chemicals were administered.

Eric M. Freedman, a law professor at Hofstra, said the Supreme Court’s impatience with arguments made by death row inmates may take a toll.

“The Supreme Court seems to be forgetting that its total supply of credibility is limited,” he said. “By consistently failing to act judiciously in death penalty cases since the spring of 2020, the court has undermined the public acceptance that it will need when it decides cases in such controversial areas as abortion and religious freedom in the spring of 2022 and beyond.”