Davis was sentenced to death in Chatham County for the murder of Savannah Police Officer Mark Allen MacPhail, The Atlanta Journal Constitution reported.
Since his trial 17 years ago, seven prosecution witnesses have recanted their testimony, the newspaper reported.
Writing for the majority, Justice Harold Melton said the court could not "disregard the jury's verdict" in the case.
"We conclude that Davis has failed to show that these alleged recantations support his extraordinary motion for new trial," the ruling said.
Melton and Justices George Carley, Harris Hines and Hugh Thompson sided with prosecutors.
Dissenting were Chief Justice Leah Ward Sears and Justices Carol Hunstein and Robert Benham.
In a dissenting opinion, Sears said she would have granted Davis a hearing to allow a judge to weigh the new evidence.
"In this case, nearly every witness who identified Davis as the shooter at trail has now disclaimed his or her ability to do so reliably," Sears wrote.