Stone skimming scientifically

In order to skim a stone it must be spinning and tilted slightly so that its leading edge is a bit higher than its trailing edge, in order that the trailing edge will hit the water first. But even though it has hit the water, the stone wants to keep spinning along the same axis, like a gyroscope. So the whole stone skips up from the surface of the water, flies a little distance, and then repeats the process. It will keep jumping as long as the stone is spinning. Gyroscopes always try to keep spinning in the same direction, so rather than flipping over when the stone hits the water, it jumps in order to keep its axis pointing in the same direction.

You can’t skim a spherical stone because there is no trailing edge to hit the water; and you can’t skim a stone that isn’t spinning because it would simply sink.