Under international law, it is legally impossible to own territory in space. You cannot claim it or purchase it. Therefor the US cannot be the owners of the Moon as it is impossible to own the Moon. They may have been the first ones there, but they have no right to it.
In order to "own" something, it is necessary to have the means to administrate, sell, transfer and such AND your ownership must be acknowledged by others. As it turns out, this is not the case of any country/person/organization and the moon. Therefore, the moon does not have an owner or owners.
In order to be considered a state, or even a nation-state, the region of argument must have a territory. The moon is considered a territory, but no one actually has direct control over it and there is no defined territory. The state must also have a population, but the moon has no people currently colonizing it. A state must also have a form of government and a code of laws (loosely defined). The moon does not have a government overseeing it and this causes it to be excluded from the next requirement, the capacity to have relations with other states. Since the moon does not have a government, it cannot declare its sovereignty, which is the final step in becoming a state.
All space-fairing nations have a "claim" on all areas of space. This is specifically for recognition purposes and is not final. Thus, no nation owns the moon, but the moon is subject for colonization by any nation with a space program (wrong flag btw).
The USA has now entered a new phase. After cutting relation with the past, this is the era of progress and conquests. After "conquering" the moon, they now want to establish a serious kingdom to rivalise the church's heaven on earth, have they not tasted too much power? .. ..