Would happen if we did not experience gravity here on earth or within our solar system?

Weightlessness.

A breakdown in the Universe as objects would move out of their orbits.

Muscle athropy and osteoporosis

Explanation:

All objects would move freely in space and also on earth. Due to this loss of mass, the human body's bones and muscles will have no weight on them hence leading to them shrinking. On a larger scale there will be no orbits to hold planets and other celestial bodies in place.