The 32-year-old suffered the injury Tuesday against the Kansas City Royals.
Gomez had been struggling at the plate, hitting just .200/.252/.345 so far this season.
In a corresponding move, Tampa recalled infielder Christian Arroyo from Triple-A Durham.
Arroyo, 22, is listed as the sixth-best prospect in the Rays organization, according to MLB Pipeline, and was acquired as part of a trade package from the San Francisco Giants for Evan Longoria in December.
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