Proves, conclusively proves beyond dispute my ignorance?! My, my, my...
OK, rather than respond in kind, I'll simply ask you what specific aspect of the Sun and Earth moving through space that *you* assert is non-Newtonian.
I done tole you and tole you: Newtonian mechanics is oversimplified for this situation; it does not apply. It must be augmented by GR, which shows that the "excess momentum" predicted by boneheaded insistence on Newtonian physics is radiated away as gravity waves.
There are numerous situations known today in which Newton is known not to suffice. In fact, the precession of the orbit of Mercury cannot be accounted for by Newtonian physics, and it was the ability of (I believe) Special Relativity to predict and account for it that confirmed that Einstein was right.
I can cite as many examples as you care to read.
If you want the full rigor, contact Dr. Carlip as I did:
--Boris