Is the whole universe is rotating on an axis?

From what we observe it is extremely unlikely the universe is rotating, but it is a good question nonetheless.

Perhaps you thought that since we see galaxies moving away from us they could be being pulled outwards as a result of the rotation. All the galaxies we observe (outside the local group) are accelerating away from us, and this includes galaxies in all parts of the universe. If the universe were like a giant spinning cylinder (or whatever object you like) then the galaxies "above" and "below" us in the cylinder would appear stationary since they are rotating at the same radius.

This observation does not support the claim that the universe is spinning on some invisible axis.

As always we should remember that our local universe may not represent the entire cosmos, and we would in fact be spinning around some axis. As more observational evidence is gathered we understand more and more about the universe, and one would be foolish to rule out a theory based on current observational evidence.