"He's a tough loss," Kingsbury said, "based on his leadership, his character, what he brings to our locker room. We're just praying for a speedy recovery."
Peters finishes the season with 15 tackles and two sacks.
Throughout the early part of this week, Kingsbury wasn't optimistic about Peters' injury before announcing the long-term prognosis a day before the Cardinals face the Seattle Seahawks on "Thursday Night Football."
Peters, who is in his sixth season with Arizona and 11th in the NFL, has established himself as a leader in the locker room and is a calming presence on the defense.
Earlier this week, before Peters' status was known, Kingsbury lauded the nose tackle, who missed the 2015 season with an Achilles injury, for his standing on the team.
"Normally it's the players trying to gain the coach's respect, but he's one of those players where, as a coach, you're trying to gain his respect because he's that well-regarded in this building and this organization," Kingsbury said. "Tremendous human being, tremendous player and leader and not a guy that you can replace in that capacity."
Arizona is now down to four defensive linemen on its depth chart, but Jordan Phillips missed Sunday's game with a hamstring injury. The Cardinals have two linemen -- Zach Allen and Rashard Lawrence -- on injured reserve, and Kingsbury doesn't expect either back for Thursday's game.
"We'll have enough guys to play," Kingsbury said.