Mind if I pick a few nits before telling you why this won't work to make you move?
Not hearing any objections, so I'll nitpick away. Note that the center of gravity (center of mass is also a better term) is a point, the axis of spin passes through it but the axis is a line, so avoid saying "the axis is this point"; it just sounds weird.
But all of that is moot because your clumsy swimming method is fundamentally flawed. You forgot one important invocation of Newton's 3rd Law. Let's run through the process.
You kick out your leg. As you correctly pointed out, the 3rd law implies that the force you use to kick out your leg also causes the rest of you to begin rotating. If you do nothing else you should keep rotating, right? Yes! If you do nothing else, you should keep rotating. But now comes the experimental part. Try kicking out your leg without doing anything else for like a minute. Bet you can't do it. Why? Because at some point, your leg can't go any farther and stops. Importantly, the act of stopping your leg means your body imparted a force on it. Just as the original force to move your leg began a rotation of your body, the force required to stop your leg cancels the rotation of your body.
The important thing to consider here is that both momentum and angular momentum must remain conserved. If you are sitting isolated in space; not moving or rotating and you don't expel or interact with any material, then your momenta are conserved. You can move or rotate your leg or arm in one direction to make the rest of your body move or rotate in the opposite direction, but once you try to bring your arm or leg back to a rest position, that cancels out the movement or rotation, leaving you with zero momentum again. What you'll find is that you'll even stay oriented in the same direction when you bring your leg back to rest position. This is because there can't be any average angular momentum over a period of time if there was none to begin with. It'd be the most frustrating clumsy swim of your life.
On Earth, there's plenty of sources of friction and other types of resistance to trick you into thinking this should work in space, but it doesn't. And if I ever become friends with an astronaut, I'll ask them to make a video during an EVA that shows how futile swimming in space is (That's gotta be a huge favour to ask of them. Space is like an enormous bottomless pit on all sides; one slip and you're gone. Can you say "agoraphobic nightmare"?)