He's the President of Nothing. The United States doesn't exist any more. He has about as much authority as the President of the USSR did in 1993.
Well technically he'd have authority over any remaining US armed forces. While regional governments after an alien invasion may not recognize the authority of a national government it would be interesting in such a scenario how individual combat units would respond to announcement that the President is alive and in control.
I've never been in the military but as I understand it they are indoctrinated to follow the chain of command and if commanders had allegiance to the President - so would they - IMO?
I'm pretty sure that there isn't any military left, either. The whole shebang was taken out in the initial strike. Anything left is just a bunch of dudes following a dude.