Can a private pilot obtain a recreational pilot certificate?
Not technically.

You could opt to surrender your private pilot certificate to the FAA thus making you not a pilot, and then you could take a checkride for recreational.

Of course, there is no reason to do this at all!  You can do the same things as long as you're exercising the privileges of the lower certificate, then whatever medical rules are there should only apply for that operation if I'm not mistaken.