In his final words, Davis maintained his innocence in the officer's killing.
The Supreme Court had rejected an eleventh-hour appeal from Davis to prevent Georgia authorities from executing him for the murder.
The court did not comment on its order late Wednesday, four hours after receiving the last-ditch request.
The filing by Davis' lawyers came after the Supreme Court of Georgia refused to grant Davis a reprieve Wednesday in the face of calls for clemency from former President Jimmy Carter, Pope Benedict XVI and others. The state parole board denied another request for clemency on Tuesday, and his offer to take a polygraph test was also rejected.
The 42 year old was set to die by lethal injection for the 1989 killing of Mark MacPhail, an off-duty Savannah police officer. Davis' execution has been stopped three times since 2007.
Supporters held vigils around the world, including in Europe, where the planned execution has drawn widespread criticism.
"They say death row; we say hell no!" a crowd of about 200 chanted Wednesday afternoon outside the Jackson prison where Davis was to be executed.
The crowd outside the prison swelled to more than 500 as night fell and a few dozen riot police stood watch. About 10 counterdemonstrators also were there, showing support for the death penalty and the family MacPhail.
Many of the roughly 150 demonstrators in Paris carried signs emblazoned with Davis' face. "Everyone who looks a little bit at the case knows that there is too much doubt to execute him," Nicolas Krameyer of Amnesty International said at the protest.
Davis' lawyers have long argued their client was a victim of mistaken identity.
Witnesses placed Davis at the crime scene and identified him as the shooter, but several of them have recanted their accounts and some jurors have said they've changed their minds about his guilt. Others have claimed a man who was with Davis that night has told people he actually shot the officer.
State and federal courts, however, have repeatedly upheld Davis' conviction. A federal judge dismissed the evidence advanced by Davis' lawyers as "largely smoke and mirrors."
The Associated Press contributed to this report.