The suspect, identified as Rhett Nelson of St. George, Utah, is also suspected in the fatal shooting of another man in central Los Angeles Monday evening, according to police.
Deputy Joseph Solano, 50, was off-duty and in civilian clothes as he stopped at the Jack in the Box in Alhambra on Monday, while getting an oil change at a nearby Jiffy Lube for a car belonging to his ill mother, police said.
Police said Solano was getting his food when a man approached and shot him in the head.
Solano was listed in grave condition and remains on life support as of Tuesday, according to Sheriff Alex Villanueva.
Officials said there appears to be no connection between Nelson and Solano and no obvious motive.
Solano is described as a dedicated family man. His father passed away a few months ago and he is now the sole provider and caregiver for his mother.
"He definitely is a kindhearted, generous person," Sheriff Alex Villanueva said. "Always willing to help others before himself. Has a bright smile, very positive individual. Everyone likes working with him, so we're all rooting for him to make a recovery."
Sheriff's officials said Nelson stopped at a church in Long Beach and called his father to say he had committed murder in Southern California.
His father called authorities in Long Beach, who were able to trace the Nelson's phone call to the church.
They found Nelson driving a short distance from the church around 10:40 a.m. Tuesday and pulled him over.
Sheriff's investigators were later able to connect him to the shooting and place him under arrest.
Nelson had previously been listed as a missing person out of Utah. In a social media posting, family members said he has a history of opioid use and had been missing since May 27.
Los Angeles Police Chief Michel Moore said investigators are also connecting Nelson to the fatal shooting of a person in central Los Angeles.
In that shooting, a car drove up to a person standing on the street. Police said the driver had a brief exchange with the victim and opened fire. The victim died at the scene. The driver fled.
The suspect, clothing and vehicle description match those in the Alhambra shooting, Moore said.
After the Alhambra shooting, a massive manhunt had been launched for the suspect seen in surveillance footage shooting the deputy at the restaurant.
The 13-year veteran of the department was in civilian clothes and waiting for his food when the suspect went up to him and shot him once in the back of the head, officials said.
"The shooting is on video," said Capt. Kent Wegener. "There doesn't appear to be an overt motive. There is no audio to the video, so we don't know if there were words exchanged."
The shooting occurred around 5:45 p.m. Monday at the Jack in the Box. The deputy had apparently gone for an oil change at a nearby Jiffy Lube and then went to the restaurant while waiting for the car service.
The deputy was taken to Los Angeles County + USC Medical Center, where a large contingent of sheriff's deputies converged at the hospital that night.
A $100,000 reward was being offered for information leading to the arrest and conviction of the gunman who shot the deputy.
The county was offering $50,000 and the Association for Los Angeles Deputy Sheriffs is matching the amount.