Personally, I don't think that a requirement like this would ultimately stand up in court. Would society demand that a restaurant owned by African Americans be required to serve racist KKK clan members dressed in their vile robes? There are plenty of establishments that can be "selective" on who they serve. ...
Ultimately, an American has a Constitutional right to exercise his/her religious faith. No law can be passed that would diminish that right. A privately owned business is owned by individuals and courts have already stated that "a sole proprietor is the same as the person."
ccchhhrrriiisss,
How about a reversal of your proposed scenario - where KKK clan members refuse to serve African Americans due to their beliefs?
In America, there is the Constitution. However, there is also legislation (e.g. Civil Rights Act, Equal Protection Clause) prohibiting racial discrimination in various contexts. It is not far-fetched to imagine these laws could be extended to prohibit 'discrimination' against same-sex couples, which is already taking place as in that news article which Proudpapa shared.
Even the First Amendment and its accompanying Free Exercise of Religion clause have been restricted in the past (c.f. Wikipedia on past judicial cases). Past legal precedents can be extended to suppress the expression of the Christian faith, so it wouldn't be too hard to restrict its practice if there is sufficient support for it in the future.
Generally, the law doesn't always work according to its literal reading, and is allowed to be re-interpreted according to the times (I guess one has to become a lawyer to know the laws he has to obey!). While I do not espouse this, this is sometimes the more pragmatic approach given that the law is replete with holes and contradictions. This applies to the American Constitution too, which was crafted by man according to the flawed ideals of the Enlightenment era.
It should be noted that in other Western countries such as the UK, Christian businesses are already coerced to provide services to same-sex couples. Fundamentally, religious rights are in conflict with a twisted notion of equality, and some courts have decided that the latter should trump the former.
But these days, businesses can have an online presence and participate in various crowdsourcing and freelancing options. So perhaps Christian businesses will find clever ways to go about this! - "Behold, I send you out as sheep in the midst of wolves; so be shrewd as serpents and innocent as doves." (Matt 10:16)