Yes, he can be made by law to that or face penalties. If he doesn't want to rent to people he might have an issue with, then he needs not open his duplex up to be rented to the public.
Really? The law of which state? Please tell us how a state law that required the things I mentioned would respect the freedom of association and the freedom of speech, both of which the First Amendment guarantees.
Are you claiming a state could make lawyers who were sole practitioners public accommodations, so that they could be penalized for declining to take every client in a protected category who wanted their services?
Are you claiming a state could make artists public accommodations, so that a painter could be penalized if he did not accept commissions from every such client--even ones who demanded to be portrayed in ways the painter considered immoral or disgusting?
How about a family with children at home whose religion considers homosexuality deeply sinful? Could the homeowners be penalized for refusing to rent a room in their house to a homosexual? Maybe the law should also require them to let the homosexual tenant share their kitchen and bathroom, or host gay parties.
How about a private book club with a couple dozen members that collects dues? What if it regularly hosts book signings at the members' homes where authors speak and sell their books, with a small percentage of the money going to the club? Wouldn't that make it a "business"? It the state law considered every business" a public accommodation, could the club be penalized for refusing to admit homosexual members?
I think the Supreme Court's decisions suggest state public accommodation laws do not have the limitless reach you seem to imagine. At some point these laws intrude so far on individual freedoms as to violate constitutional rights.
Two decisions in particular discuss both the expressive and intimate subtypes of the freedom of association, and how that freedom applies to state public accommodation laws--Roberts v. United States Jaycees and Boy Scouts of America v. Dale.
The freedom of speech applies to all sorts of creative expression, and it includes an individual's freedom not to be compelled by government to express or sanction ideas he does not agree with. Supreme Court decisions on government-compelled speech include West Virginia Board of Education v. Barnette, Wooley v. Maynard, and Pruneyard Shopping Center v. Robins, in particular what Justice Powell said in his concurring opinion.