Padres manager Bud Black made the announcement Saturday after a 3-all tie with the Chicago White Sox.
The oft-injured Johnson was projected to be the team's fourth starter. He figured to pitch the first game of a three-game series against the Miami Marlins on April 4.
Johnson, who spent the first eight years of his career with the Marlins, led the NL with a 2.30 ERA in 2010. He signed an $8 million, one-year contract with San Diego after going 2-8 with a 6.20 ERA in 16 starts for Toronto last year.
Black announced the injury to Johnson after saying Ian Kennedy and Tyson Ross will follow No. 1 starter Andrew Cashner in the rotation.