I believe it does not.

Here's an easy experiment. Go to a nearby tree. Make sure it's not toxic in any way, and grab a small branch with several leaves or pine-needles on it and soak it in your hot bath water until the water is cool enough to take a bath in.

Just join the branch with the bath water, scrub real good, use soap, and filter the bathwater when it goes down the drain with a washcloth or scotch-brite. After it all goes down, remove the branch. What do you see at the bottom of the tub?

Notice any difference in the itchiness of your scalp or the rest of your body?

I theorize that the fluoride is effected somehow by the green leaves, perhaps bound up or made inert somehow. It allows the skin to get cleaner, the toxins to be expelled better, and dead skin drops off easily.

See for yourself and report the results.