In a corresponding move, Arizona promoted quarterback Trace McSorley from the practice squad to the 53-man roster.
"It's unfortunate," Cardinals coach Kliff Kingsbury said. "But we'll get Trace going and keep it rolling."
McSorley will take over McCoy's role as Kyler Murray's backup. McCoy, who turned 36 on Monday, wasn't on the field for the open portion of Monday's practice.
"Obviously, it's unfortunate, kind of starting the week off on a downer with somebody getting hurt," Murray said. "Obviously, somebody you care about, on his birthday -- it's tough. That was tough but, no, Trace is good. Trace is obviously ready to go."
The Cardinals also added quarterback Jarrett Guarantano, who was with Arizona throughout training camp, to the practice squad.
McCoy appeared in eight games and started three while Murray was sidelined with an ankle injury last season, completing 68 of 90 passes for 684 yards, three touchdowns and one interception. The Cardinals went 2-1 in his three starts.
McSorley, who was drafted in the sixth round in 2019 by the Baltimore Ravens, has played in three games but hasn't logged a start. He didn't get into a game last season.