Question: Consider the electric field surrounding an electron. The electron moves. A wave propagates at $c$ reflecting the electron's movement. This is an electromagnetic wave. Consider a mass. Classically surrounded by a gravitational field. The mass moves. The field deforms and it propagates at $c$. 

* Is this not a gravitational wave?
* Why do people talk as if only general relativity predicted gravitational waves?

Answer: 
Newtonian gravity doesn't predict anything traveling at speed $c$ as neither the constant $c$ nor any constants you can use to make the constant $c$ even appear in Newtonian Gravity.

Maxwell has $\epsilon\_0$ and $\mu\_0$ from which you can make $c$. And General Relativity and Special Relativity have $c$ directly as a constant of the theory. But Newtonian Gravity has no speed, it only has one constant, $G,$ and the units of $G$ are $\mathrm{kg^{-2}N}.$
You're not wrong! Gravitational waves *do* occur classically, and that's because general relativity is a classical theory! In Newtonian gravity, the gravity field doesn't "deform," nor does it have a speed - it's just a force that acts at a distance, instantaneously. So you're entirely correct - that is one way that gravitational waves could be created.