"It might be possible to end delays by limiting constitutional protections for prisoners on death row," wrote Justice Stephen Breyer. "But to do so would require us to pay too high a constitutional price."
Bucklew's execution has been delayed several times, and now-retired Justice Anthony Kennedy had joined liberals in the last stay.
But Kennedy has been replaced by Justice Brett Kavanaugh, who voted with Gorsuch and the rest of the court's conservatives to let the execution move forward.
Missouri plans to use an injection of a single drug, pentobarbital, to carry out Bucklew's execution. But he suffers from a congenital and rare disease called cavernous hemangioma. It causes blood-filled tumours to grow in his head, neck and throat, which his lawyers say could rupture during the state's lethal injection process.
Breyer said Bucklew had developed persuasive evidence that lethal injection could cause him to "sputter, choke, and suffocate on his own blood for up to several minutes before he dies."
That would subject him to "constitutionally impermissible suffering," Breyer wrote. "The majority holds that the state may execute him anyway. In my view, that holding violates the clear command of the Eighth Amendment. "
Breyer was joined by Justices Ruth Bader Ginsburg, Sonia Sotomayor and Elena Kagan.
Gorsuch and Kavanaugh made up the majority along with Chief Justice John Roberts and Justices Clarence Thomas and Samuel Alito.
In 1996, Bucklew, now 50, stalked his former girlfriend Stephanie Ray at another man's trailer. He shot and killed the man, Michael Sanders, tried to shoot Ray's fleeing child and then captured Ray. He handcuffed and raped her, then wounded a police officer in a subsequent gunfight.
Bucklew later escaped from jail and attacked Ray's mother with a hammer before he was recaptured.
He was not challenging the constitutionality of the death penalty, or even of Missouri's lethal-injection procedure. His claim is that, applied to his specific condition, the protocol would result in the kind of tortured death the Constitution forbids.