In corresponding roster moves, the team recalled outfielder Mason Williams and right-hander Chris Martin from Triple A Scranton/Wilkes-Barre.
An MRI on Miller's forearm on Wednesday revealed the injury. Miller will not throw a baseball for the next 10-14 days, after which he will be reevaluated.
"I think it was just one of those things that has kind of been building and I've been doing everything I can to try and make sure I'm available and capable, and it just hasn't been recovering quite like I want it to," Miller said, according to MLB.com. "It's frustrating more than anything for me. I want to be out there and contribute, and you feel like you're letting guys down and you can't go out there and do your job."
Miller has excelled for the Yankees during the 2015 season, striking out 43 batters in 26 innings and logging a 1.03 ERA.
He has 17 saves on the season, which is tied for second in the American League.
The Yankees could turn to Miller's fellow bullpen star, 27-year-old Dellin Betances, to step into the closer role. Betances has allowed one earned run in 32 1/3 innings so far this season.
New York signed Miller to a four-year, $36 million contract in December, and he took over Yankees' closing duties from David Robertson during the 2015 season.
The 30-year-old lefthander converted from a starting pitcher to a relief pitcher in 2012 when he played with the Boston Red Sox.
- Will Green and Scooby Axson