We absolutely can't, with current technology, move an asteroid of any dangerous size towards (or away from) a collision with the Earth - and even more so not with a single engine that has to have fuel for the entire duration of the flight. It would take thousands of years to get an ion engine on an asteroid to move it back to the Earth, and you'd have to constantly send fuel with it. And controlling where it will land is an entirely different problem.
And many years of the engine running would still be easily reverted by one or two high energy impacts, like a fusion bomb hitting the asteroid just right.
Consider this as well from an energy diff perspective - to hit a city with the same energy as a nuclear bomb, you have to have the final mass times the final speed squared about equal to the energy of the fusion bomb. And to get that, you have to put in energy that's proportional to the mass before burning up in the atmosphere times the square of the delta of the speed difference between its original speed and its speed on a collision course with the Earth. So you need to find an asteroid that's pretty close to hitting the Earth, and still put in the energy to move its full mass the extra distance.
You're right this is not fully a symmetrical problem, but it's asymmetrical in the opposite direction: defense is a lot easier than offense in this case, simply because of the massive inertia of the weapon. To alter your metaphor of the bullets, there's a reason why we shoot explosive rockets instead of just hurling huge rocks at our enemies. And it's precisely the same reason why a nuclear weapon is much better than an asteroid.
And many years of the engine running would still be easily reverted by one or two high energy impacts, like a fusion bomb hitting the asteroid just right.
Consider this as well from an energy diff perspective - to hit a city with the same energy as a nuclear bomb, you have to have the final mass times the final speed squared about equal to the energy of the fusion bomb. And to get that, you have to put in energy that's proportional to the mass before burning up in the atmosphere times the square of the delta of the speed difference between its original speed and its speed on a collision course with the Earth. So you need to find an asteroid that's pretty close to hitting the Earth, and still put in the energy to move its full mass the extra distance.
You're right this is not fully a symmetrical problem, but it's asymmetrical in the opposite direction: defense is a lot easier than offense in this case, simply because of the massive inertia of the weapon. To alter your metaphor of the bullets, there's a reason why we shoot explosive rockets instead of just hurling huge rocks at our enemies. And it's precisely the same reason why a nuclear weapon is much better than an asteroid.