Is gravity responsible for the rotating of a planet on its axis?

Answer 1

Not directly. Barring any collisions in the past with other planets, the rotation of a planet is the result of the rotation of the protoplanetary disk.

The planets formed out of a disk of gas that was circling around the newly formed Sun. When the gas clumped together into the different planets the so-called conservation of angular momentum ensured that the planets rotated as well. This is also the reason planets all rotate in the same direction. Of course collisions between protoplanets in the past may have altered the axis of several planets (Uranus being a prime example).

Gravity is indirectly responsible for the rate of rotation; it was the force that clumped the gas together thereby speeding up the rate of said rotation. This is again the principle of conservation of angular momentum; if a thing becomes smaller it rotates faster with the same angular momentum.

You can test this yourself by sitting in an office chair (one that can freely rotate), and having someone giving you a whirl. If you extend your arms you will slow down, while if you retract them you will speed up (it works even better if you carry weights in your hands). It is also the mechanism whereby an ballerina speeds up in a pirouette.

Answer 2

No. A planet rotates on its axis because it has angular momentum.