Remember, a theory can never be verified, only falsified. Mach's principle is only poorly incorporated in Einstein's theory. You enter a rocket, and you take a journey in the universe along a certain path. Whether our Universe is rotating is a question for measurements, not theory. The outcome of measurements can be either 'Yes, it's rotating', or they can put an upper limit to the rotation, if any.
Measurements can never prove that the rotation is exactly zero.
As it is, there is no certain evidence that our Universe rotates. Let's have a look at the upper limits then. The positions of the planets are in practice always determined in relation to the background of the fixed stars. One of the parameters is the rotation of the background. It turns out that the latter parameter is zero within the experimental errors.