Within days of being sentenced to life imprisonment with a 21-year minimum non-parole period, Bracken appealed the conviction and sentence, but that was turned down by the Court of Appeal in March.
He then took a case to the Supreme Court asking for the right to appeal. In a decision released this week, the country's highest court dismissed his application for leave to appeal.
Supreme Court judges Justices Susan Glazebrook, Sir Terence Arnold and Mark O'Regan said in their decision it was not in the interests of justice to grant the application for leave to appeal.
During the trial, the jury heard how 30-year-old Mr Davis was kidnapped and "hog tied" before being driven into remote bush where he was murdered.
Justice Edwin Wylie said during sentencing: "It was calculated and vicious. It was, in effect, an execution."
The judge said Bracken was "dangerous and evil".