If accelerating reference frames can be treated as normal inertial reference frames but with gravity, then for a rotating reference frame, is the centrifugal force = gravity? More specifically, I was reading about Mach's Principle as it was called by Einstein. When we spin and feel our arms fly, is it due to gravity caused by the rotational acceleration?
No, it is due to centrifugal force. Even if you go to deep space, where there is no heavy body around that would cause gravity, and spin, you will still see/feel your arms fly. I am not sure though how you will be able to spin yourself in deep space.
But then theories consider gravity and any other acceleration equivalent. In that sense, one can call it gravity but I think it will be a stretch.