The President of the USA is the commander in chief of the U.S. armed forces according to the Constitution of the USA.
Therefore, he can deploy troops and withdraw troops as he sees fit. Of course, preferably with Congressional approval. Congress can always kill the funding, as they did in 1975 when they refused to send aid to South Vietnam, resulting in North Vietnam overrunning South Vietnam, going against President Ford's wishes.
However, plenty of examples where Congress was not in the mix for troop/naval/air deployments other than Iraq (1990-91, 2003-present) and Vietnam:
Balkans-1990's
Somalia-1992
Panama-1989
Grenada-1983
Mayaguez rescue-1975
Korea-1950
Since WW2, the days, it seems, of declaring war is over, as the USA has not declared war on any country, and only Congress can do that. That is what happens when you have agreements and treaties with organizations such as the U.N., NATO, etc. where use of force is authorized under certain conditions.