Centripetal force is not a force that exists in its own right. In order for an object to remain in circular motion, there needs to be a force ALREADY pulling the object towards the centre of the circle to PROVIDE centripetal force.
For example, in swinging a yoyo in a circle, the force of tension provides centripetal force. When a car drives around a sharp curve, the force of friction provides centripetal force. And when a person stands on the Earth, the force of gravity provides centripetal force.
Therefore, if Earth’s gravity turned off, a person would simply continue to follow his tangential velocity and fly off into space. (Of course, the whole planet would have disintegrated as well, so you might not notice.)