Officials said they didn't have a motive for Routh's attack. The three men apparently traveled to the gun range together in the same truck.
Authorities said they tracked Routh to his home in Lancaster, Texas, on Saturday evening, where police tried to persuade him to turn himself in. Instead, he made a break for it, and police gave chase.
He was arrested without a fight after officers spiked his tires on a freeway, Bryant said. A handgun was found in his home, authorities said. They wouldn't comment on whether he had any other weapons.
Routh was being held in lieu of $3-million bail and didn't have an attorney yet, officials said.
A recent sampling of 1,388 Iraq and Afghanistan war veterans by the University of North Carolina-Chapel Hill School of Medicine reported that one-third had shown aggression toward others in the last year, with 11% reporting they'd used or threatened to use a knife or gun against another person, gotten into a fight or tried to rape someone.
PTSD, homelessness, substance abuse and joblessness were listed as risk factors.
In Kyle's book, he hinted at the struggles he faced when combat slowed down: mentally replaying the times he'd been shot, brooding over his mortality, which he pushed away during combat. He said doctors put him on drugs to help him cope with the psychological stress he faced after his fighting days were over.
Kyle, who was married with two children, also recalled the tension his wife felt over his deployments: "If you die, it will wreck all our lives," he quoted her as saying, adding that she was furious that "you would not only willingly risk your life, but risk ours too."
But he wrote of being at peace with his work and his faith.
"When I die, God is going to hold me accountable for everything I've done on earth," Kyle wrote at the close of his book, adding: "But in that backroom or whatever it is when God confronts me with my sins, I do not believe any of the kills I had during the war will be among them. Everyone I shot was evil. I had good cause on every shot. They all deserved to die."
Pearce reported from Los Angeles and Hennessy-Fiske from Texas. Times staff writer Tony Perry in San Diego contributed to this report.