The general concensus seems to be that the President has the authority but that he would still need to turn to Congress for the funding. (I seem to recall a treaty that prohibits nukes in space. I doubt it would provide much of a constraint given the situation, but it might make testing problematic.) I don't believe for a minute a "black ops" account exists that is large enough to fund a project as huge as this.
The unsuitability of ballistic missiles to deliver the warheads is a good point. Am I wrong in assuming that NASA has and would use another type of rocket?
Some of the responses seem to suggest that trying to deflect an object with a nuclear weapon would be futile. But wouldn't that really depend on the size of the rock, the size and/or number of the nukes (after all, we've got thousands of 'em), the composition, and, maybe most importantly, the distance to intercept?
I think it makes sense to be prepared to do what we can do. It would be a shame if we were not ready to give a nuclear nudge to a 1/4 km wide comet that comes along because we got hung up trying to figure out the best way to shove a 5 km wide iron-core asteroid that isn't. By all means we should develop other methods but until then we should think about how best to use the tools we have.
And we definately need to get an early-warning system in place.