Actually, the smaller object exerts the same force on the larger object.

('Same' meaning equal but opposite, of course.)

The gravitational force exerted on body1 by body2 is determined entirely by both their masses, the universal constant G, and the inverse of the square of the distance between them.

It's totally indpendent of which object is bigger (and also independent of which is more massive, if you want to get nitpicky about terms).

For similar and derived laws, see Kepler's Laws, or Newtons.