I'mma be honest here: I'm horrible at math.
I see the formulas but I don't quite see how it all goes together.
physics is a son of a *****. thanks for the headache.

it could be explained holistically, i hope...
G (capital g) is representative of a constant, so the masses of the objects express that constant on eachother. remember, not only a rock falling, but newton's third has it that the earth is rising toward the rock. not only does the earth hold the moon in orbit, but the moon pulls the tides higher, representative of the reciprocal force.
Shouldn't, then, a more massive object 'fall' at a greater rate? Isn't this increased gravitational force from a more massive object the very reason we 'fall' on the moon more quickly than on earth?
the third law is the equal and opposite axiom, so you can appreciate that gravity hasn't escaped the greater implications of conservation. you were pointed in the right direction with inertia. given conservation, could energy be expressed both in velocity by way of acceleration,
and mass by way of inertia? rather, the effective mass of a 5 pound brick is > 5lbs when dropped from height. in a vacuum, it will drop at the same rate as a feather.
as to the moon, our 5lb brick registers lighter on a scale on the moon. now, knowing that gravitation is constant, and that the mass of the brick is the same, plus the idea that there is equal and opposite force expressed between the brick and the planet it is on, the indication is that it's the moon's mass that's responsible for the change in the brick's weight. the difference between the equations on each planet is conserved in inertia, and this propensity for the brick to stay put, afloat is indicated by its slower attraction to moon's surface.