It absolutely does not. Newtonian gravity occurs instantly. It has no notion of information taking time to propagate. But we know gravitational waves happen, so Newtonian gravity is wrong _at even very large scales_. If the sun disappeared Newton tells us we'd know immediately. In GR we'd know about 8 min later.

The bigger problem is not that the quantitative effect is large, but that the _qualitative_ difference of going from the instantaneous effect to one that needs to propagate is enormous. It's the whole point of relativity as a concept.

Even going to GEM as a true, non-singular linear approximation of GR would be a step up from Newton's laws, at least there we can have gravitational waves and causal flow of information.