McCoy, starting in place of the injured Kyler Murray, who tore an ACL in a Monday night loss to the New England Patriots, went down after a 1-yard run. He got up on his hands and knees but fell back to the ground.
After McCoy was evaluated on the field by the Cardinals' medical staff, a cart drove onto the turf to bring him to the locker room, but he refused the ride and walked to the locker room on his own.
Arizona's third-string quarterback, Trace McSorley, replaced McCoy, who was 13-of-21 with 78 yards and an interception.
Coach Kliff Kingsbury said after the loss that McCoy will be in the concussion protocol.
"I think he's doing OK," Kingsbury said. "You never like to see that happen. It's a tough impact, going out there seeing him like that but, talking to our trainers, I think he's doing better, so that's good."
McSorley said seeing McCoy go down "was tough" to watch.
"Everyone's praying for him, feeling for him and hoping he has a speedy recovery," said McSorley, who was 7-of-15 with 95 yards and two interceptions in the game.
Of the two other quarterbacks on the team, David Blough was inactive for Sunday's game. He was signed to the active roster, and quarterback Carson Strong was signed to the practice squad on Wednesday.