Roger Wayne McGowen was condemned in 1987 for fatally shooting a 67-year-old woman during a Houston bar robbery in March 1986.
The 5th U.S. Circuit Court of Appeals decision upholds a lower court's ruling that instructions given to jurors didn't provide a way for them to consider his disadvantaged background.
The ruling posted late Monday is in line with other Texas capital cases of that era before the courts refined jury instructions.
The appeals court also rejected arguments that the 48-year-old McGowen is innocent, his trial lawyers were deficient and his long time on death row is unconstitutionally cruel and unusual.