Actually, part of his job IS to determine what is or is not Constitutional. (That is, it could be a good basis to veto a bill, for example.)
His problem is: he has no fucking clue about what qualifies something as Constitutional or unConstitutional.
That's the Courts role, you know that. He can Veto Anything he doesn't like, just because he doesn't agree with it. That is within his power. His Power is mostly Executive, Commander in Chief, and Presidential Order. For Each, he is accountable. Here, he is putting himself above the Court and many times over, Due Process and Oversight. He has decreed that the Rule of Law is what he says it is. His terms are a fail and a threat.
I feel queasy saying this, but Liability had it right. The President takes an oath to "preserve, protect and defend" the Constitution. It is implicit to the furtherance of that responsibility that the President make determinations about how to interpret provisions. This is why the outrage over Bush's use of signing statements was overblown to some extent (though it was still an issue in cases where he simply said he'd ignore the plain text of the law).