While such a little push might work there is too much to go wrong. I prefer the single high yield event of a ground penetrating warhead with a very high yield.

I'm not a physicist or geologist (a biologist actually), but from what I've read so far, the gentle nudge approach, started in advance, is the best way to do it. By contrast, an explosive device meant to shatter the asteroid would be far worse, leading to multiple large meteors raining down on the earth.

re; Oliver Medvedik
The penetrator nuke is not intended to shatter the asteroid it is to add reaction mass to the push on the asteroid but if you shatter the asteroid all the pieces have been accelerated onto different courses; was not this the plan from the start.

How ironic would it be if the "paintball" method pushed it into the gravitational keyhole?

Sounds overcomplicated and retarded. Who's to say that the paint would stick, "dry" or not pick up dull space dust. A large meteor would have a small gravitational force and would have a haze of find dust and debris around it. the object would most likely be spinning and of course moving. Even Also the paintballs could be to hard from the cold or heat up too much in the sun, the release or firing mechanism would be the obvious point of failure. Targeting would also be tough.
The solution that most experts have agreed on is the one stated correctly by Pikeman. You use explosive to break off a chunk of the rock and thus push the large mass.

When I saw the article title, I imagined a completely different scenario. In the scenario I imagined, the momentum of the paintballs would be added to the momentum of the asteroid. In the scenario described, it looks like that momentum is proposed to be effectively neutralized prior to arrival with relatively slow speed paintballs not significantly affecting the momentum.

Explosions don't work. Nuclear explosions are too fast to be able to effect the huge mass, inertia, and kinetic energy. Breaking off a chuck will not change the trajectory of either chunk. If you are going to try something, this is much better than nuclear explosives. But the best is to attach mass thrusters that dig into the asteroid and accelerate and eject reaction material. That is much more expensive however, but has the advantage of being possible to control over time.

re; Rigby5
A rocket is a device throws off mass in one direction to accelerate other mass in the opposite direction.
If you use an above surface detonation the only reaction mass you have is the device used and any mass you vaporized from the surface and much of this is wasted by being allowed to expand indirections that do not push the mass you are trying to move. This is why it is optimal to use a subsurface detonation in which everything thrown out of the crater including still solid rock is reaction mass in a short lived but very powerful rocket burn.
Your best is exactly what a subsurface detonation nuke does.

As I calculate it, the gravitational force, though small, would be enough to grab any matter (paint) within 100 meters of the surface in a minute or so. Adhesion might be a problem, but there would be plenty of time for multiple applications to get it right. If it is required to prevent disaster and does so, the value would clearly be immeasurable.

I think paint balls would merely create a dust cloud as the pellets splat on the surface. Once the dust resettles, the paint will be covered.
The best way to permanently get rid of an asteroid is to decay its orbit into the sun (or Jupiter). I wonder how much of the asteroid's makeup is water or other frozen liquids and gasses. Landing a reactor on the asteroid then melting off the ice could reduce its mass, thus it easier to get pulled into the Sun.