Without the force of gravity from the Earth, it would just float away into space. This mix of velocity and distance from the Earth allows the Moon to always be in balance between fall and escape. If it was faster, it would escape; any slower and it would fall.
Also - A missing moon could cause even more disruptive changes, although on a much longer time scale. Without the moon's gravity holding the Earth in place, the tilt of our home planet's axis would probably shift drastically over time It has no large, stabilizing moon to stop it
