Kenneth Williams was the fourth inmate put to death by the state of Arkansas in the past eight days after a dozen years without a single execution.
His attorneys said witness descriptions of the lethal injection procedure were “horrifying”.
The United Nations on Friday condemned Arkansas’ execution schedule.
“I see no reason for any investigation other than the routine review that is done after every execution,” Governor Asa Hutchinson responded during a Friday press conference.
The state had initially planned to put to death eight men in a span of 11 days, but courts issued reprieves in half those cases.
The compressed schedule was because of the state’s stock of one lethal injection drug, the sedative midazolam, was due to expire at the end of this month, said officials.