Than why would object A with a large mass "fall" to the Earth with the same acceleration as object B with a small mass?
What is the cause?
Consider a body of mass m and a planet of mass M. Hold the body of mass m at a height above the body of mass M. Let go.
The force of gravity on mass m is:
F=GMm/r^2.
where r is the distance between the centres of the two masses.
But Newton's second law tells us that acceleration is proportional to the force applied and inversely proportional to mass, specifically
a=F/m.
Rearranging:
F=ma.
Equating the two equations for F we get
ma=GMm/r^2.
Cancelling the m's we get
a=GM/r^2.
That is, a is independent of m.