O'Brien was convicted in February of leaving the scene of an accident that killed 43-year-old pedestrian Jim Reed. The bishop had until Thursday to file an appeal.
Superior Court Judge Stephen A. Gerst sentenced O'Brien to four years' probation and ordered him to spend 1,000 hours of community service ministering to the sick and dying.
O'Brien, who grew up in Indianapolis and attended St. Meinrad Seminary in southern Indiana before he was ordained a priest in 1961, resigned as head of the Phoenix Diocese within days of the fatal accident. He's believed to be the first Roman Catholic bishop in U.S. history to be convicted of a felony.
"I accept both the verdict of the jury and the sentence of Judge Gerst and accordingly, will not be filing an appeal in this case," O'Brien said in a statement. "I feel relieved that we have come this far in this whole unfortunate, sad and exhausting ordeal."
O'Brien, 68, also expressed appreciation to his supporters and "my deep regret to the Reed family over the great loss that they have sustained."
Reed was drunk and jaywalking the night of June 14 when O'Brien hit him. The crash left the passenger's side of O'Brien's windshield badly cracked.
The bishop testified that he heard a loud crash but never saw anyone in the road. The defense argued that dim lighting, headlight glare and the victim's dark clothes made Reed hard to see, but prosecutors argued O'Brien knew or should have know he hit a person.