Inmate Kenneth Biros had waited for the decision hours past his 10 a.m. scheduled execution time at Ohio's death house.
Prisons director Terry Collins said the execution would not happen Tuesday.
The execution team had been waiting in a holding pattern while the court decided, ready to administer the lethal injection if the court had granted to the state's request to go ahead with the execution.
The justices' one-sentence decision agreed with two lower courts that had ruled to delay the execution, including the 6th U.S. Circuit Court of Appeals that refused earlier Tuesday to allow a hearing before the full court to consider a state appeal.