If the Machians are right, then of course rotation of the universe is impossible. I write about Mach's principle as it is pretty well known among scientists - but possibly it is best known among amateur scientists. Even Lenin had opinions about Mach's principle. However, Mach's principle has no place in modern science since it explains absolutely nothing. He formed the General Relativity Theory much along these lines.
Matter tells space-time how to curve.
The galaxies tell the space-time how to curve through the field equations. In particular, space-time tells what is rotating and what is not. The General Relativity is the best theory that we have today for the large scale structure of the universe. Remember, a theory can never be verified, only falsified. Mach's principle is only poorly incorporated in Einstein's theory.