How gently could a comet/asteroid/meteorite "hit" Earth?
Yes, spacecraft do it all the time by using air resistance (and sometimes rockets) to slow down.
Meteorites enter the atmosphere at high speed, typically 10-70 km/sec, but the smaller ones are slowed by air resistance, so they typically hit the surface at just a few hundred kilometers per hour. Reference: http://csep10.phys.utk.edu/astr161/lect/meteors/impacts.html (that link is now dead).
As gerrit points out in comments, a sufficiently small micrometeroid, like a dust particle, probably could hit the surface gently.
Larger bodies are not affected nearly as much by the atmosphere. For any such body, the velocity at impact is going to be at least several kilometers per second. A body moving obliquely in the same direction as the Earth's rotation would have a slightly lower speed at impact, but the impact still could not be described as "gentle"; the Earth's rotational speed is still a small fraction of orbital or escape velocity.
Here's one way to think of it. Freefall trajectories are reversible. If you watch a movie of an incoming body in reverse, it still makes physical sense (ignoring air resistance). Any meteorite hitting the surface has to have been in deep space at some point before the impact. If there were a trajectory that allowed such a body to have a "gentle" impact speed, then it would be possible to start with the same body near the surface at the same "gentle" speed, but in the opposite direction, and have it reach deep space. Unless the meteorite has its own propulsion system, that's just not going to happen.
You can get to space with a lower starting speed by moving along with the Earth's rotation - which is why most rockets are launched to the east, to take advantage of that. Reversing such a trajectory can result in a slightly slower impact, but only slightly.
Depends on the origin of the asteroid too, interstellar asteroids? No. Trojan asteroids perturbed just right to hit earth at the perfect, low-grade hyperbolic trajectory that dips below the edge of the atmosphere (without going to low)? Maybe.
@MagicOctopusUrn: No, that's still not possible. For a body big enough that air resistance doesn't have much effect, it can't hit the Earth's surface at low speed. Remember, it still has to fall from the top of the Earth's gravity well to the bottom of it. Try dropping or throwing a rock from the top of the Empire State building and having it hit the ground gently.
Your reverse trajectory argument is correct, thus the lowest (most gentle) impact speed is escape velocity from earths I.e. 11.2 km/s.