In 1944, a committee of the House known as the Woodring Committee held hearings on the subject—just before the end of the war. Most people date unification discussions from that period. But it will interest you to know that in the preceding twenty-five years there were about thirty suggestions and bills introduced in Congress looking to unification of the army and the navy.
"The Historical Evolution of Our National Security Organization,"
Naval War College Review: Vol. 7
, Article 2.
Available at: https://digital-commons.usnwc.edu/nwc-review/vol7/iss1/2