WASHINGTON — The Supreme Court heard arguments on Tuesday in an appeal from a death row inmate in Missouri with a rare medical condition that he says will cause excruciating pain if he is put to death by lethal injection. Lawyers for the inmate, Russell Bucklew, said his condition, cavernous hemangioma, would make him choke on his own blood during his execution.

It was Justice Brett M. Kavanaugh’s first death penalty case, and there is good reason to think that he holds the crucial vote. In March, five justices voted to stay Mr. Bucklew’s execution. Justice Anthony M. Kennedy, whom Justice Kavanaugh replaced, joined the court’s four more liberal members to form a majority; the court’s four more conservative justices were in dissent.

Much of Tuesday’s argument concerned earlier Supreme Court decisions that required inmates challenging lethal injection protocols to identify available and preferable methods of execution. Mr. Bucklew said lethal gas was preferable to the state’s current method of an injection of a lethal dose of pentobarbital. But the United States Court of Appeals for the Eighth Circuit, in St. Louis, ruled that Mr. Bucklew had not shown that his alternative would be less painful.

Justice Kavanaugh seemed to express doubts about the requirement of identifying an alternative, at least where the usual method of execution coupled with an inmate’s unusual medical condition could produce excruciating pain.