Einstein did not “debunk” or “replace” Newtonian gravity. He refined Newtonian gravity.

There are not two forms of gravity. There is one. When the velocity of the objects is not significant and when not near a large gravitational body, use the basic formula: f=GMm/r^2

When the velocity is significant or near a large gravitational mass use Einstein’s field equations.

If you want to use Einstein’s field equations for low velocity and not near a large gravitational mass, you can. The answer will be the same to many digits of precision.

Does a rotating sphere as suggested by Ernst Mach, induce Coriolis force?

Einstein’s Theory of Relativity and Mach’s Principle

An application of Einstein’s equation to test the hypothesis that a rotating sphere will induce Coriolis and centrifugal forces.

Abstract:

The equations of motion of a test particle near the center of a rotating spherical shell with the mass M and the radius R are investigated in the framework of E~nstein’s theory of relativity up to the post-post-Newtonian order of approximation. Among the forces acting on the test particle, the Coriolis and the centrifugal forces appear. In order that Mach’s thought about rotation is realized, two conditions on M/R must be imposed. It is shown that these two conditions are not consistent with each other.

Conclusion: “We conclude that Mach’s thought is not realized, in the case of the rotating spherical shell up to the post-post-Newtonian order of approximation. This conclusion will be unchanged in higher order of approximation.”