Depends on timing. ICBMs can only reach an altitude of about 1300km and are not designed to hit moving targets in space anyways, so if an asteroid is going to h

Depends on timing. ICBMs can only reach an altitude of about 1300km and are not designed to hit moving targets in space anyways, so if an asteroid is going to hit us in the next few days we're screwed. However implementing a system to defend against asteroids even with short notice is feasible. We can make rockets much larger than ICBMs and put them on standby for planetary defense. You'd presumably put appropriate sensors and maneuvering thrusters for getting to the approrpiate position near or on the target, akin to the interceptors used in missile defense. Interceptors would have a dedicated 3rd stage like the Centaur to launch them on trajectories away from Earth - the further away the interception the smaller the necessary deflection. Ideally you'd catch it early and start the deflection so far away that a kinetic impactor or even just the craft's gravitation would nudge it to a safe course, but for short notice you need to go nuclear. Breaking up an asteroid into smaller pieces with a nuke is possible, especially for the subset of asteroids which are basically loosely held together piles of gravel. Some people think this is a bad idea as instead of 1 big rock to deal with now you have lots of little rocks that are also radioactive, but that actually is a much preferable situation. The ability of objects to pass through the atmosphere is dependent on their diameter, lots of small rocks will burn up harmlessly while a single big one will impact. The radioactivity from the nuke would be spread out extremely dilute, and specific nuclear weapon designs which minimize neutron activation can keep the radioactivity very low. A more serious concern is that you don't know the trajectories of the resulting fragments, some of which might still be dangerous, and thus it would be a very bad idea to destroy it before impact was a certainty, but then waiting until the last minute to stop it is a major risk in itself. More safe would be to detonate the nuclear device near the asteroid. This would vaporize a thin layer of the surface of the asteroid in line of sight of the explosion, sending it flying off into space at extremely high velocity and propelling the asteroid equally in the opposite direction. There are two major advantages to this approach: it will work against targets regardless of their composition and internal structure (very beneficial if we don't have time to study the asteroid ahead of time) and the method can be done repeatedly to the same target - we could launch dozens of independent interceptors and just keep detonating them one by one until we get the desired result. While the goal is to have the asteroid not impact at all, and thus radioactivity is even less of a concern, this option does produce even less radioactivity than blowing up the asteroid. From a strictly technical perspective this option is pretty much a no brainer, but obviously building a battery of dozens of nuclear missiles is politically problematic.