Still less could we send up a shipload of space cowboys to do the job for us, as in the movie Armageddon; we no longer possess a rocket powerful enough to send humans even as far as the Moon. The last rocket that could, Saturn 5, was retired years ago and has never been replaced. Nor could we quickly build a new one because, amazingly, the plans for Saturn launchers were destroyed as part of a NASA housecleaning exercise.
Even if we did manage somehow to get a warhead to the asteroid and blasted it to pieces, the chances are that we would simply turn it into a string of rocks that would slam into us one after the other in the manner of Comet Shoemaker-Levy on Jupiter—but with the difference that now the rocks would be intensely radioactive. Tom Gehrels, an asteroid hunter at the University of Arizona, thinks that even a year's warning would probably be insufficient to take appropriate action. The greater likelihood, however, is that we wouldn't see any object— even a comet—until it was about six months away, which would be much too late. Shoemaker-Levy 9 had been orbiting Jupiter in a fairly conspicuous manner since 1929, but it took over half a century before anyone noticed.
Interestingly, because these things are so difficult to compute and must incorporate such a significant margin of error, even if we knew an object was heading our way we wouldn't know until nearly the end—the last couple of weeks anyway—whether collision was certain.