The same will happen as with other abandoned colonies and exploration missions of the past: it will just disappear into the dust of history.
As to “51st state”, I think Newt meant it not as taking posession of the moon for the US so much as the way it would be administered (as an independent political entity rather than a scientific or military station directly under federal control, or some sort of corporate governance by the establishing companies along the lines of the diamond mining operations deBeers sets up in southern Africa).
This of course implies it would be a rather large operation, large enough to have its own political leadership, law enforcement agencies, etc..





