The_Jazz

If there are no inherent flaws in the rock, then yes, that would work just fine, assuming that you detonated close enough with enough force.

The inherent problem with that idea is that the rock may not be able to withstand the shockwave and could shatter. Remember that these pebbles have been bouncing around for 4 billion years, and the impacts are going to cause cracks and faults. If you set off a warhead too close, you could just end up shattering the whole thing but not pushing it into a new orbit since the force intended to move it ended up being absorbed in the fracture process. That means that instead of 1 large body hitting, you've got hundreds or thousands of small to medium sized bodies. The small ones would probably only cause local damage, but the medium sized ones can pack a big wallop. The Great Crater in AZ was caused by a boxcar-sized rock that impacted and left a hole a mile across and spread large debris 20 miles. If you have 20 of those hitting around the world, the destruction could actually be greater than what would have happened if the single body had impacted. Not that it would matter since human civilization would pretty much be over under either circumstances.

The real problem is that we don't have a good way of gauging what flaws are inherent to any large body. What looks like a solid rock could actually be a collection of gravel held together by their own gravity. From millions of miles away, it's awful hard to tell which it is. The size and distance of the explosion are going to need to be determined by what it is that's on track to impact. The missile option is our best bet, but it's by no means 100% sure of removing the threat.

However, you should probably be more concerned about being outside during a thunderstorm than the earth being hit by an asteroid. The odds are much better for the lightening.