Actually, gravity does not cause objects to rotate. It is their angular momentum which does this. As a figure skater pulls in her arms, analogous to gravity pulling on two bodies, her spin increases. This happens because angular momentum must be conserved, not because the skater has a particular mass.
As for the universe, some cosmological models predict that the universe could have a net spin, and this would be detectable in the isotropy of the cosmic background radiation as a 'quadrupole' type anisotropy. The current limits to any such spin are now so restrictive that, to the best of our measurements, the amount of spin in the universe is so small that it would have had no significant cosmological consequences. It amounts to a rotation speed of less that 10^-12 radians per year or 10^-7 seconds of arc per year!