Abbott used strong language to outline Medellin's crimes, which include participating in the gang rape and strangulation of two young teenage girls, Jennifer Ertman and Elizabeth Pena.
"Medellin was tried and convicted of murder during the course of a sexual assault," Abbott wrote. "A jury unanimously recommended a death sentence."
Texas had asked the high court to consider the presidential memorandum to be a "request," not a "command."
Government lawyers wrote in legal papers supporting Medellin that the president "has recognized authority to resolve disputes with foreign nations over individual claims, and to establish binding rules of decision that preempt contrary state law."
The government lawyers said that the "compelling national interests served by the president's determination outweigh the relatively modest intrusion on state interests."