newton theory of gravition and at which point it invalid?

Answer 1

Newton's theory of gravitation is valid for most general cases that we see and observe.

It may not be so accurate when dealing with sub atomic particles or particles with very high mass density (like neutron star, black hole etc) which actually distorts the space.

Answer 2

It depends at which degree of accuracy you are doing your calculations.

Earth's orbital speed is approximately 0.0001 c (c = speed of light). At that speed, calculations involving lengths, apparent momentum, apparent time, etc., will be off by approximately 1 part in 2,000,000,000.

If you use Newtonian instead of Relativistic, the length of the year will be off by almost 1/50 of a second.

Can you tolerate such a difference? (I can)

The idea is that the difference exists as soon as there is (relative) motion. For some calculations, you have to worry even at very slow speed (e.g., calculating the speed at which electrons propagate in an electrical wire -- which is quite slow). For others, you can still make some use of Newton's formulas even at almost half the speed of light.

In general, though, if you are in a situation where you find it easier to express the speed in fractions of the speed of light, then you should use Relativity instead of Newtonian.