Newton said the instant the Sun were to just go away the planets would continue in the direction they were headed at the time,
More to the point, Newton's law of gravity assumed that gravity acted instantaneously across space.
Einstein first theory of gravity didn't do that as light takes 8 minutes to reach the Earth. 10 years later Einstein refined his theory saying gravity acts at exactly at the speed of light, which satisfied Newton's laws.
Einstein didn't have a "first" theory of gravity. His 1905 paper on special relativity had nothing to do with gravity, but it did postulate that no physical effect (including gravity) can propagate faster than light. Ten years later, he finished his general theory, which was indeed about gravity.
In Einstein's general theory, gravity acts at the speed of light, but that doesn't "satisfy" Newton's assumption of instantaneous action. Einstein's general theory does approach Newton's law for the case of smaller masses and shorter distances. And Newton's law can be "patched" to accommodate the speed-of-light propagation of gravity, but not the other effects in Einstein's general theory, such as large-mass corrections, space-time effects, bending of light by gravity, and so on.
cite: PBS The Elegant Universe - Einsteins Dream
I think you might want to watch the show again. BTW, this thread is about radio waves not gravity.