Einstein with his assistants, Infeld and Hoffmann, calculated the first two stages of this approximation and found that in the first stage the equations of motion take the Newtonian form (Einstein, Infeld and Hoffmann 1938, 65 - 66). In this approximation, if we consider very low accelerations then the exact equations of motion indeed take the Newtonian form and we obtain a material particle that cannot radiate. In this state of affairs, we have revived the good old assumption that there could be no gravitational waves.
Einstein first proposed non-existence of gravity waves in 1936, then recanted in 1937 when an error in his calculations was discovered, and then reinstated his proofs of no gravity waves in 1938 as the above referenced material indicates. Now unless some one has a later dated paper by Einstein showing his 1938 paper (Einstein, Infeld and Hoffmann 1938, 65 - 66) [en.wikipedia.org] to be in error, as it stands at present, Einstein clearly believed there is no such thing as gravitational waves.