Is there an Abraham-Lorentz force for Gravity? The Abraham-Lorentz force in Electromagnetism is the recoil a charge experiences as it accelerates due to own emission of radiation.  
The Newtonian theory of gravity is identical to that of electrostatics.  Admittedly, Newtonian gravity doesn't have an analogue for the magnetism half of EM (as far as I'm aware), so I can't necessarily prove the existence of waves / analyse radiation / derive Abraham-Lorentz.  
However, modern gravity does have propagating waves in it.  Do accelerating masses have an Abram-Lorentz-like force?
 A: Yes. Here is one investigation of the gravitational back-reaction:
https://arxiv.org/pdf/0907.0412.pdf
However, the most common case where a gravitational back-reaction is important (and actually observed!) is a two-body system like the Hulse-Taylor binary. In this kind of situation, you can calculate the rate at which energy, momentum and angular momentum are radiated away in gravitational waves, and this is relatively easy to do analytically in a post-Newtonian approximation. Then, by conservation of these quantities, you can assume that the two-body system is losing energy, momentum, and angular momentum at that rate. I think this is much more common than trying to calculate the effect of the back-reaction forces on each of the two bodies.
The important thing is that we have observational evidence for gravitational back-reaction. The Hulse-Taylor binary, like other binaries, is inspiraling at exactly the rate expected. And the LIGO events are consistent with back-reaction... the back-reaction is what caused the black holes to spiral together and merge.
