Problems with Newton's Law of Gravitation:

1.

It gave the wrong prediction for the precession of Mercury's orbit

2.

It didn't explain why the gravitational force on an object was proportional to its inertial mass (i.e. why objects fell with an acceleration independent of their mass and composition).

3.

It was inconsistent with Einstein's Theory of Special Relativity: if an instantaneous force of attraction existed between two distant objects, information about the location of one is instantaneously transmitted to the other (i.e faster than the speed of light)