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?