It seems unlikely to me that there exists a way to count actual fixated mutations and classify them as to whether it was due to selection or drift, so I don't think we could know either way whether what you're claiming is true. Maybe one of the biologists will chime in.
Population genetics gives us a maximum on fixation by selection, since selection involves killing/inhibiting reproduction of individuals. So it can only select for so much in a given generation, all the rest is hope to genetic drift.
We do know that advantageous alleles will be driven toward fixation more rapidly than neutral alleles, but as fixation approaches the effects of drift become more prominent and can actually interfere with fixation. Apparently, in sexual populations recessive advantageous alleles can actually have a better chance of fixation than dominate advantageous alleles simply because of the effects of drift. See Natural Selection, Genetic Drift, and Gene Flow Do Not Act in Isolation in Natural Populations.
And if genetic drift has such on a strong effect on the fixation of mutations, wouldn't one be justified to question if such an amount of randomness can still account for the evolution of complex structures ?