There is absolutely a gravitational radiation reaction and solving for it is one of the very active fields in classical relativity theory at present. Basically, particles with nontrivial masses distort the spacetime around them; this causes them to not move on geodesics of the "background" spacetime (the spacetime that one would have found had the secondary particle not been there) but instead on some accelerated path. Solving for that path "self-consistently" at all orders is the holy grail of this field. This turns out to be very difficult.
Presently, the so-called "self-force" is understood to first order in the Schwarzschild spacetime. It will perhaps soon be understood to second order there and to first order in Kerr. The calculations are very challenging.
The self-force is of somewhat more practical interest than the electromagnetic radiation reaction because it is responsible for the inspiral and ultimate merger of binary black holes, which is the most important source of gravitational radiation. It turns out to have two components: a "conservative" one which respects the symmetries of the background spacetime, and a "dissipative" one which roughly accounts for the energy lost to gravitational radiation.
Understanding both is necessary to model gravitational wave sources, particularly in the "extreme high mass ratio limit" where $M_1 ~10^7$ solar masses and $M_2 ~ 1$ solar mass. This should realistically describe the case of, say, a stellar mass black hole merging with one of the supermassive black holes at the centre of a galaxy.
Eric Poisson, The gravitational self-force is a review. Peter Zimmerman, Eric Poisson, Gravitational self-force in nonvacuum spacetimes and Eric Poisson, Adam Pound, Ian Vega, The motion of point particles in curved spacetime provide details.
EDIT: The gravitational self-force is not responsible for inertia. It is a very weak effect normally.