Circular motion can be thought of as analogous in many ways to linear motion. Consider that you are on the surface of Earth, which is rotating. Therefore you are also rotating, however relative to Earth, it makes sense to say that you are not rotating. Similarly with the moon. If it were rotating relative to Earth, then we would expect to be able to see what is on its current dark side, as it would be the lit side after half a rotation.

Also, a rotating object does not need to produce or receive any force. The conservation of angular momentum says that unless a force acts on a system, it will continue to rotate at whatever angular velocity it is currently rotating at. This is because the system itself is not accelerating. If you look at a particlar subsection of the system (say at an end), then that subsection is accelerating due to a centrifugal force, but that force is contained within the system.