He is expected to miss a minimum of four weeks, the team said.
Gallo had left Tuesday's game against the Mariners and returned to Texas for further testing.
"Really disappointed but I'm actually not surprised after what I saw the last four games, especially the swing," Rangers manager Chris Woodward said. "I think it's been bothering him for a little bit."
Gallo has 22 homers and 49 RBIs for the Rangers, who were contending for an American League wild-card spot at the beginning of the month but have now have fallen 6½ games back.
The 25-year-old outfielder also missed time in June with a strained oblique injury.
The hamate is a wedge-shaped bone on the outside area of the wrist, on the same side as the pinkie finger.
Gallo was placed on the 10-day injured list retroactive to Wednesday. Outfielder Willie Calhoun was recalled from Triple-A Nashville to take his spot on the roster.
In another set of moves, the Rangers acquired right-hander Shane Carle from the Atlanta Braves for cash and optioned him to Nashville. To make room for Carle on the 40-man roster, right-hander Kyle Dowdy was reinstated from the 10-day IL and designated for assignment.
Dowdy was on a rehab assignment at Nashville. He was placed on the injured list June 1 with a right elbow impingement.
Carle was 4-2 with a 5.13 ERA in 20 games, including one start, at Triple-A Gwinnett in the Braves organization. He was designated for assignment Wednesday by Atlanta.
Carle spent last season in Atlanta's bullpen, going 4-1 with a 2.86 ERA in 53 appearances.
The Associated Press contributed to this report.