Sikh Americans have been battling mostly unsuccessfully for years to serve in the U.S. military while keeping their turbans and beards—important elements of their South Asian religious faith.

Now, they have won some new allies: a group of 27 retired military leaders, including a retired three-star U.S. Army general, have signed a letter that an advocacy group called the Sikh Coalition plans to deliver Wednesday to Defense Secretary Ash Carter, saying Sikh Americans “should be given equal opportunity to serve.”

“The...