Oh dear.

Professional cleaning, maybe? Could there be something nasty in the links, maybe a chemical of some sort, that works its way out when you're wearing it, but is wiped away from the surface when you give it a clean?

Also, if it's very hot where you are, could it be a heat rash where the watch is effectively blocking sweat glands, like prickly heat? I'd try another watch for a day or two, see what happens.

Failing that, it is possible to be come sensitised to something and end up with an allergic reaction. If it's nickel, rhodium plating can help.

I can't think of anything else to suggest. I really hope you can sort this, because a rolex is a lovely, lovely thing.

If you can bear to be without a watch (or do like DW does normally and wear it on her right wrist), I'd leave the sensitised wrist without anything rubbing or blocking it for quite a few days.

If it is nickel allergy, unfortunately it will crop up on the other wrist too pretty soon. If it isn't, leaving the skin to recover from a topical dermatitis for quite a few days will maximise chances of recovery/non-recurrence

Uggh...thank you everyone.  I was really hoping I could rhodium plate it and everything would be good to go. 
Guess I need a plat Rolex.   

I will try everyone suggestions, one by one.