A Gay Web Designer is not under obligation to design a custom site for the Westboro Baptist Church, are they?
Probably not, but It really depends on two things.
1. If the business is open to the public. (i.e. It comes under the legal definition of a Public Accommodation.)
I doubt if Web Design, by its very nature would qualify. This is more applicable to restaurants, hotels and other businesses where the doors are literally open to the public.
2. Whether his/her work qualifies as a personal and/or artistic expression.
The courts have ruled in multiple cases now that cakes and floral arrangements in and of themselves (sans any sort of a message, emblem, picture or slogan) do not fall into this category.