Actually, there is a somewhat paranoid group of righties out there pushing this exact scenario.
However, the Supreme Court has made it very clear that a treaty cannot be used to override the Constitution, as is of course obvious on its face to anyone but an idiot.
According to SCOTUS, the order of authority is as follows.
Constitution
Laws and treaties of the US
State laws
That said could the UN Treaty trump state laws on say concealed carry?