I had the same problem with the back of my watch, which was stainless steel, BTW, making a rash on the back of my wrist. I have a solid gold watch now and I don't have any problems with it at all even though it's not 24 karat gold, which I think would be too soft to use in jewelry.
I think that white gold has some nickel in it as well, so you would need to wear yellow gold if you're allergic to nickel.