The right to not be discriminated against due to your race, religion, gender, or sexual orientation. Your question is too broad in that you seem to be intentionally avoiding any mention of protected class, as though you are unaware that this issue isn't about "refusing to do private business with someone," it's about refusing to do private business with a person solely due to their race, religion, gender, or sexual orientation. Clearly, the Civil Rights Act is a precedent here, and I would suggest that what you are fundamentally opposed to is that act, or at least parts of it. If we hadn't passed the Civil Rights Act, what do you think America would look like today? What would have happened during the lifetime of MLKJ? Now please consider what it is that you are endorsing.
Since the businesses who have made the news in relation to the issue of discriminating against homosexuals have cited religious freedom, let me offer a suggestion. How about a bakery is allowed to refuse business to homosexuals as long as it also refuses businesses to all sinners. Anybody who has had a divorce and anybody who has had sex before marriage. Otherwise, a bakery who serves the latter groups of people but cites religious freedom to justify denying service to the former is apparently trying to find a legal justification for their bigotry.
