Is there a Newton-like equation for gravity that gives equivalent results to that of general relatively?

Physicists have tried simple extensions to Newtonian gravity to make it compatible with Special Relativity. For example, Nordström’s 1912 theory replaced the Poisson equation relating the Newtonian gravitational potential $varphi$ to the mass density $rho$,

$$nabla^2varphi=4pi Grho,$$

with the obvious special-relativistic generalization,

$$nabla^2varphi-frac{1}{c^2}frac{partial^2varphi}{partial t^2}=4pi Grho.$$

This is a wave-like equation. It predicted gravitational waves, pre-Einstein!

Unfortunately, this theory does not agree with observation, nor do other simple extensions of Newtonian gravity.

Going in the other direction, one can start with General Relativity, consider the problem of $N$ gravitating masses in the case where their gravitational fields are weak and their velocities slow (such as for planets in a solar system), and make their equations of motion look like Newtonian gravity plus various small corrections. The result is known as the Einstein-Infeld-Hoffmann equations.

They are a kind of series expansion of General Relativity in powers of $1/c$, where the lowest order terms with no $1/c$ are the Newtonian theory. The expansion has been carried to several orders higher than the $1/c^2$ corrections shown in Wikipedia, so it can be extremely accurate.

These equations are integrated to make planetary ephemerides, such as for interplanetary spacecraft navigation. It is much easier to deal with these equations than the field and geodesic equations of General Relativity.

Physicists could never have guessed the Einstein-Infeld-Hoffmann equations. Since they are a series expansion, they are not simple and elegant. They can only be derived by starting from General Relativity, which, in its own way, is just as simple and elegant as Newton’s theory.