He was arrested on Monday after the investigation of a hit-and-run-incident which killed Jim Reed, 43, led the police to the bishop's car, which had its windscreen caved in and the bumper dented.
Fr O'Brien, 67, who had recently made an agreement to avoid a possible indictment for sheltering child abusers, told the police that he thought he had hit a dog or a cat or that someone had thrown a stone at his car.
The Pope accepted his resignation in a two-line announcement yesterday citing the clause in church law which allows a cleric to resign for illness or "some other grave reason" that makes him "unsuited for the fulfilment of his office".
The police are investigating whether drugs or alcohol were involved. On Tuesday Fr O'Brien was charged with leaving the scene of an accident involving an injury, which carries a sentence ranging from probation to three years nine months in prison.
If he is found to have been driving while intoxicated the charge could be increased to negligent homicide or manslaughter.
"Somebody died," the Maricopa county attorney, Richard Romley, said. "There has to be accountability somewhere."
Fr O'Brien was released on $45,000 bail and the Archbishop of Santa Fé, the Most Rev Michael Sheehan, has been appointed interim administrator of the diocese.
Earlier this week the chairman of the panel monitoring the church's efforts to resolve the child abuse crisis, the former Oklahoma governor Frank Keating, resigned after comparing bishops to the Cosa Nostra in their willingness to cover up wrongdoing.