I would advise against it. It may be law that a person can't be fired for that direct reason, but it can color management's opinion and it can have indirect repercussions. It can be used against you because people have perceptions that influence their decisions indirectly.
Unless you are personal friends, friends outside of work, I just would not provide any information about any diagnosis. If you think your medication might affect your work, I would just advise your supervisor that you will be beginning a new medication and you might need time to adjust to it. That is all that you need to share.