Supreme Court guidance says mental illness can't disqualify someone from execution if they understand the sentence and reasons for the punishment, the state lawyers argued.
Green had declined to speak with reporters as his execution date neared.
Investigators questioned Green at least twice in the days following Christina's disappearance 12 years ago. His wallet was found in some woods near clothing and jewelry that belonged to Christina, but authorities found nothing else of significance at the time. A few weeks later, a tip from a neighbor about an unusually large burn pile behind his ramshackle home brought them back again.
While Green had been cooperative in the past, he grew testy and ordered them off his property when an FBI agent looking at the fire site detected the smell of a decaying body and inserted a metal probe into a patch of disturbed earth. They returned hours later with a search warrant and a dog trained to detect human remains.
The dog led officers to the girl's body, stuffed inside a laundry bag in the home and wedged into a corner behind a piece of furniture. Green contended someone else had placed the body there and that he was being set up.
Evidence at his trial indicated he had tried to burn the body, buried it in a shallow grave, then removed it when detectives left to obtain the search warrant. DNA from her remains tied him to the slaying. A carpet fiber from her panties found in the woods was traced to a carpet in his home.
Two years ago, Green came within about four hours of execution before the Texas Court of Criminal Appeals stopped the punishment amid similar arguments he was too delusional and too mentally ill to be put to death.