WASHINGTON—U.S. Defense Secretary Ash Carter on Wednesday said Army Gen. Mark Milley would be nominated as the next Army chief of staff and that Navy Adm. John Richardson will serve as the next chief of naval operations.

Gen. Milley, a former commander in both Afghanistan and Iraq, was the senior officer who recommended charges against Sgt. Bowe Bergdahl, the Army soldier held captive by the Taliban for five years before he was released last year. Gen. Milley also served as the Army’s base commander at Fort Hood, Texas during...