Rhett McKenzie Nelson, 30, of St. George, Utah, was arrested Tuesday in the shooting of sheriff's Deputy Joseph Gilbert Solano at a Jack in the Box in Alhambra.
He was initially held on $2 million bail, but that was changed to no bail following the deputy's death.
Nelson had been expected to be arraigned Thursday, but the proceeding was postponed until July 22.
The District Attorney's Office announced Thursday that Nelson has been charged with two fatal shootings, one attempted murder and two counts of second-degree robbery.
The other fatal shooting happened in central Los Angeles only hours before the deputy was shot. Officials say Nelson was in his vehicle when he approached a man on the street, exchanged a few words and then shot him. That victim has been identified as Dmitry Koltsov.
He also is accused of attempting to murder a man named Aleksandr Chudetckii.
Hours after shooting Solano, officials say Nelson also committed two armed robberies in Long Beach - one at a Shell gas station and the other at a 7-Eleven.
If convicted as charged, Nelson faces a possible maximum sentence of death or life in prison without the possibility of parole. Prosecutors will decide later whether to seek the death penalty.
He is also suspected in a string of armed robberies of convenience stores in the San Diego area.
Solano, 50, was shot in the head as he was waiting for food at a Jack in the Box on Monday.
Nelson was reported missing by his family as of May 27. In a Facebook post, family members said he had a history of opioid abuse, but has been clean for about six months.
Nelson was caught thanks to a tip to police from his father.
Investigators say Nelson was at a church in Long Beach when he called his father in Utah and told him he had been involved in homicide in Southern California. His father contacted Long Beach police and they were able to track the phone number and find him near the church.
Detectives say he was driving the white Kia Sorento matching the description of the vehicle used after the deputy's shooting.
Investigators say a gun matching the one used in the Alhambra shooting was found in Nelson's car.