"The weather was warming up and I feel like I was kind of making the adjustments I had been looking to make," he said. "It's just unfortunate this kind of stalled what I thought was going to be a really nice run."
Miller struck out 17 and walked four over 10 scoreless innings in 11 games this year.
"We lean on him so hard," Indians manager Terry Francona said. "Then there's the temptation to bring him back early. Even though he doesn't want to go on the DL, it's the right thing to do. It's just hard to do it. But it's the right thing to do."
Miller doesn't think he'll be out longer than the 10 days.
"I think I won't have to shut it down for too long," he said. "Hopefully I'm back on a mound pretty soon, and I can get back to maintaining that direction."
Miller was on the disabled list with a left hamstring strain while with Boston from March 26 to May 5, 2012, and missed four games with a left hamstring strain while with Baltimore in September 2014.
A two-time All-Star, the 32-year-old left-hander was on the DL twice last season with patella tendinitis in his right knee. Miller was acquired from the New York Yankees on July 31, 2016, for four minor leaguers.
Left-hander Tyler Olson replaced Miller on Wednesday and got Rizzo to fly out before retiring two batters in the eighth.
"That was really impressive," Francona said. "You kind of feel punched in the stomach when Andrew has to leave the game. And then if things unravel from there, it makes for a really tough night. But the way he pitched, it made it kind of fun."