Considering the President did not actually end up uttering the precise words needed for the swearing-in (as prescribed by the Constitution), there are bound to be some kooks who will claim that he is not actually president (a nutty proposition, of course, but a potentially annoying one - like those "he was born in Kenya/Indonesia/Mars rumors). My prescription to the problem? Swear him in again. It's actually happened before:
People will argue about what the failure to utter the words in the precise order required by the Constitution means. But it will be an academic argument. It is not clear who would have standing to raise the argument that Mr. Obama had not become president as a consequence, and it is hard to believe that any court – or other body – would want to adjudicate the question.For the record, this is the only part of the Obama presidency that I'd be comfortable modeling after Calvin Coolidge.
There is, in any event, no rule against a do-over. When questions were raised about whether Calvin Coolidge should have been sworn in by his father, a notary public, he took the oath again, this time from a Supreme Court justice.