What centripetal force acts on particles of a rotating planet (for example)? Any particle going in circles has a centripetal force acting on it, then what force acts on particles of a freely rotating object?
For example, particles at the surface of the Earth (or any particles of Earth) complete one revolution about the North-South axis every 23Hr56Min, what force acts on them?
It is not gravity for sure, as there is no relation between an object's mass and rotation speed.
Rotating objects are not considered inertial.
 A: 
It is not gravity for sure, as there is no relation between an object's mass and rotation speed

But... it is gravity. The gravitational force on particles is related to their rotational speed around the planet. I don't see why your claim about the mass of the object would disprove that.
Rotational speed $\omega$ relates to the tangential (linear) speed $v$ via a geometric bond:
$$v=\omega r\,$$
with $r$ being the distance from the centre of the circular path (so, from the centre of Earth).
A centripetal force causes a centripetal acceleration related to the tangential speed, and by substituting in the above geometric bond, you arrive at:
$$a_c=\frac{v^2}r=\omega^2 r\,.$$
When gravity is the cause of the centripetal acceleration, then via Newton's 2nd law we now have a direct relation:
$$F_g=ma_c=m\omega^2r\,.$$
It is not given that gravity is the cause of the centripetal acceleration. It could be any force. And it often is a mix of forces, to be honest - in this example we are idealizing away many other forces that might tweek the exact behaviour (the exact centripetal acceleration). But in general for any circular orbits around Earth's centre, gravity is basically the only candidate that at every point is perpendicular (centripetal) and thus very often has a huge part to play, if not in practice the only significant part to play. Consider, say, a satellite at the edge of atmospheric influence as a "clean" example without disturbances by other forces. There, only gravity interacts. So what else than gravity could be causing the rotational motion?
