Eddie Ray Routh, 27, is accused of fatally shooting Kyle and Kyle's friend Chad Littlefield multiple times at a gun range about 110km southwest of Fort Worth in February 2013 and then fleeing in Kyle's pickup truck.
Defence attorney are trying to have Routh declared innocent by reason of insanity and called a psychiatrist who testified he is a paranoid schizophrenic and showed signs of psychosis that could not be faked.
Prosecutors, seeking a life sentence without the possibility of parole, contend Routh knew what he was doing and acted deliberately when he shot the two and then fled. They called a psychologist who testified Routh has a personality disorder make worse by heavy drug use and has been faking schizophrenia.
Rebuttal witnesses were expected to take the stand on Tuesday, which would then be followed by closing arguments. After that, the jury of 10 woman and two men at the court in the rural city of Stephenville would begin deliberations.
The trial has focused renewed attention on Kyle, who is credited with the most confirmed kills of an U.S. military sniper, and on the movie "American Sniper."
Routh, who served with the Marines in Iraq and Haiti, had been admitted to Veterans Affairs hospitals after his service to treat mental illness. Prosecutors said Routh did not see combat in his overseas deployments.
Routh's attorneys have not disputed that he shot Kyle and Littlefield. In videotape provided by police and shown in court in the rural Texas city of Stephenville, Routh admits to the shooting in a rambling speech that defense attorneys contend attests to his unstable mental state.
The trial was halted on Monday due to an ice storm in the area.
