SteveW

If the object is free falling in a vaccum then the earth applies gravitational force against the object. The object also applies gravitational force against the earth to pull the earth towards it. These forces are equal and oppositely directed. Since the earth is so much more massive, it resists the force applied to it by the object and effectively doesn't even move where the motion of the object free falling is significant.

The earth's motion is determined by all the forces acting on it. The sun applies gravitational force on the earth and so do other planets and all of these forces determine the net external force acting on the earth.