Specifically, Article II of The Outer Space Treaty declares:
“Outer space, including the moon and other celestial bodies, is not subject to national appropriation by claim of sovereignty, by means of use or occupation, or by any other means.”
Other provisions include requiring the use of Mars for peaceful purposes, consistent with international law. America is also prohibited from establishing “military bases, installations and fortifications, the testing of any type of weapons and the conduct of military manoeuvres on celestial bodies…” (Article IV)
In short, America may have gotten there first, but all we get are bragging rights.