Not quite. As the ball falls it is following a ballistic orbital trajectory, and for an object with no other forces acting on it this will indeed be following the path that a great circle would describe. It will not be falling along a ground path on a line of latitude - in the northern hemisphere it will always turn to the south, and of course vice-versa in the southern hemisphere. A great circle can't be a line of latitude, except at the equator. All lines of longitude are great circles./QUOTE]
I still don't think that a ball falling vertically describes a great circle as it isn't passing between two points on the surface of a sphere. It's travelling down the plane of a great circle but, IMO, it's not passing along the arc. As I said, a great circle can be a line of latitude or longitude but, as you point out, there's more of one than the other!