Count me as *** for the 'apology' route.

I'd take it a step further however... I would write the apology letter to your direct supervisor and CC it to the boss whose permission you DID get. It would read the same as the letter by COURTMOUNT outlined above, with 1 small addition.

"In the future, I would ask that any discussions on my behavior / attire be dealt with in a professional manner, instead of berating me in front of my peers and the store's customers. Such circumstances are best dealt with in private as it's unprofessional to accuse someone of being a 'tranny' in public."

I would apologise to him for your alledged 'unprofessionalism', as well as call him out for HIS unprofessional behavior.