Can someone who know about Georgia's judicial system explain me why all the appeals were rejected ? I mean, aren't such high doubts about his guilt a sufficient ground to suspend the procedure ?
What doubts? State had 34 witnesses, including numerous friends and acquaintances. Later some of his homies decided to retract the testimonies to save him from the needle. Post-trial retractions are widely considered laughable as "proofs" of innocence, especially when there are dozens of witnesses who are not retracting.
And by the way the other victim was black just like Davis, just like the majority of jury that sentenced him to death in less than two hours. He lost the SCOTUS appeal 9-0, and never found one single judge in 20 years of trying that would believe his "ONE-ARMED MAN DID IT!!!" nonsense.
There weren't "dozens of witnesses". There were 9, of whom 7 recanted, one of whom confessed to the crime himself (not something you're likely to do to protect "your homie"). Take your racism elsewhere.
The fact of the matter is that jury convictions, even incredibly unjust jury convictions, are extraordinarily difficult to overturn, to the point of absurdity. The standard of overturning a jury conviction is literal "proof of innocence", which is just plain impossible for the vast majority of innocent convicts. The Supreme Court made their decision 9-0 because the standard of "proof of innocence" is impossible to meet. There may have been twenty years of appeals, but none of those appeals really stood much chance of exonerating the innocent. A big part of the problem is the irrational respect for jury verdicts.
(In answer the original question, declare a mistrial and start all over--he might well have been guilty, but it certainly was never proven beyond a reasonable doubt. Also, no death penalty regardless, obviously.)