NATO, relying on US assets in Europe, could have US airborne forces deployed in key transit sites within about 3 hours, which is how such an operation would likely go, with a secret request from Ukraine announced only afterward.  Russia would not be able to react in time to a US deployment without actually having to risk shooting US troops already in place, in which case it would be "game over" for Putin (and potentially for a lot more than him), just as it would have been "game over" for Ukrainians if anyone had shot Russian troops in Crimea (or, vice versa, if Russians had shot Ukrainian troops). That's the gamble of that strategy.  

Neither the US nor Russia has ever, historically, seriously relied on legal mandates for anything determined important enough to use military forces, so legal issues just aren't a serious factor here. Laws are a discursive, diplomatic tools, not a actual restrictive impediments, at the level of world powers like the US and Russia.

Putin risked the possibility of nuclear war in Crimea already, as well as the lives of his undefended and unmarked troops. That means, strategically, using simple Prisoner's dilemma game theory, only a similar willingness to risk such a thing on the part of the US can secure Ukraine from Russian tanks. Because Ukraine really isn't very important to the US, however, it's unlikely that Obama would sign off on that gamble, and all appearances are that the US would rather risk losing at least the eastern part of the Ukraine in an effort to satisfy Putin's needs, at least being able to claim the partial success of having installed a government hostile to Russia in Kiev, which was a major defeat for Putin from which he partially recovered with the invasion of Crimea. A unified Ukraine is less important to NATO, and Russia, than a buffer state Ukraine at this point, so I don't expect to see NATO forces deployed there, and I do expect to Russian forces entering eastern Ukraine at some point if the current set of Ukrainian Ukrainian leaders remain in Kiev.