McCann, 35, left Tuesday's 5-1 win over the Miami Marlins in the third inning with the injury.
He is hitting .264 with 10 home runs and 39 RBIs in 74 games this season. He also was on the IL in April due to a right hamstring strain.
The Braves recalled catcher Alex Jackson from Triple-A Gwinnett to fill the open roster spot. Jackson, 23, is hitless in three games and 10 at-bats for Atlanta this season.