The Rangers will place him on the disabled list before Thursday's game with the Chicago White Sox and make a corresponding roster move.
"It stinks. It's frustrating," Hamilton said. "You work your tail off to get here, the guys are doing well when I got here, you're able to have fun with them, it's frustrating. There is nothing I can do except work hard to get better. I have had success before getting back early. Hopefully I can do the same."
Said Rangers manager Jeff Bannister: "I feel bad for Josh. We had a lineup before he got here. We'll continue to use that lineup."
Hamilton suffered the injury Sunday after touching first base on his pinch-hit, walk-off double in a 4-3 victory over Boston at Globe Life Park.
"I kept wrapping it every day in [Double-A] Frisco and [Triple-A] Round Rock," Hamilton said Tuesday. "On Sunday, I didn't wrap it.
"I promise you, I thought, 'You need to wrap it,'" Hamilton said. "'No, I'll be all right.' I hit the ball, and as I rounded first, I felt it grab. I took four more steps and felt it grab again."
Hamilton, acquired from the Angels on April 27, played in 12 games in the minor leagues before joining the Rangers in Cleveland on May 25. He played in seven games with the Rangers.
Hamstring may sideline Hamilton up to a month
Jean Jacques-Taylor discusses Rangers outfielder Josh Hamilton's hamstring injury, and how he may have been impacted by a lack of a full spring training.