Gomez, 31, suffered the injury while running the bases during Sunday's game against the Oakland Athletics.
He is batting .246 with four home runs and 13 RBIs in 164 plate appearances this season.
The Rangers have not yet announced a corresponding move to fill Gomez's spot on the roster. Jurickson Profar, Jared Hoying and Drew Robinson are the most likely options to fill the void, according to the Fort Worth Star-Telegram.
Delino DeShields is expected to move from left field to center field while Gomez is sidelined, with Ryan Rua playing left field.