Jamal Darwish Fatayri was executed on Monday after being sentenced to death for shooting dead Naeb Omran Ma'ayteh, a first secretary at the Jordanian embassy in Lebanon.
Fatayri - who was linked to the former Palestinian fighter Abu Nidal - was hanged at Swaqa prison south of Amman.
"The trial of Jamal Darwish Fatayri fell seriously short of international standards and claims that 'confessions' had been extracted under torture were ignored," Amnesty said in a press statement.
The organisation called on Jordan to ban the death penalty and to hold an independent inquiry into Fatayri's "unfair" trial.
Claims of torture made by Fatayri and others who have either been executed or sentenced to death must also be investigated, Amnesty said.
Jordan has executed more than a dozen people since last year.