I think you're right about the general shape of the curve; however, I don't think the ball would make it all the way to the "center" of the spiral. Reason: resistance to rotation and resistance to tangential motion. Any physical ball has an irregular surface upon which molecules of air will impinge, always resisting the direction of motion. Since the "top" part of the ball will be moving faster relative to the air than the "bottom" part of the ball, the resistance to rotation will be greater there, hence reducing the rotation. And of course, with 1 atmosphere of air present, there will be air resistance pushing against the tangential velocity of the ball. So I think you might start in on your spiral, but I don't think you'd get to the center.