You may, as a Christian, believe homosexuality is a sin. And you may, as a Christian, marshal biblical and theological arguments to support that belief. I know those texts, I know those arguments, and I understand the resulting theological opinion. [Just to be clear, I do disagree with it, however: I believe that gays and lesbians are sinners in exactly the same way heterosexuals are sinners--by virtue of our common sinful human nature, not by virtue of whom they love.]
However, what is entirely illegitimate is to take that belief one step further, and argue, on religious grounds, your right not to serve gays and lesbians. This is what has happened in Arizona: read the story here--http://www.cnn.com/2014/02/21/us/arizona-anti-gay-bill/
The opening line of the story says it all: "Arizona's Legislature has passed a controversial bill that would allow business owners, as long as they assert their religious beliefs, to deny service to gay and lesbian customers."
Here's the problem. While Jesus didn't say anything explicitly about gays and lesbians, he did have a great deal to say about one's neighbor and the obligation a Christian has in regards to her neighbor. In a word, that obligation is love. Let's take just two of the most famous examples. First, in the Gospel of Matthew, a lawyer asks Jesus, “Teacher, which commandment in the law is the greatest?” Jesus replies, “You shall love the Lord your God with all your heart, and with all your soul and with all your mind. This is the greatest and first commandment. And a second is like it: you shall love your neighbor as yourself. On these two commandments hang all the law and the prophets” [Matthew 22:36-40]. The second example comes from the Gospel of John, in the hours before Jesus’ arrest.
After Jesus has washed his disciples’ feet and is preparing for his betrayal and death, he says to his disciples, “I give you a new commandment, that you love one another. Just as I have loved you, you also should love one another. By this everyone will know that you are my disciples, if you have love for one another.”
These two verses make clear that love is a primary (if not the primary) mode of being in which Christians are to live their lives: it suffices as a summary of all the other commandments and prophetic words of the Lord; it is an indispensable core component of fidelity to God; and it is the defining characteristic of one’s identity as a follower of Jesus. Elizabeth Johnson writes that since God is present in the whole of the world, “profoundly present and committed to the world and every person in it,” “loving God means loving the world.” Thus, it can be argued persuasively that it is impossible to love God without loving one’s neighbor, and that calling oneself a Christian while hating others is a fundamental contradiction. And, let's be clear about Jesus' understanding of "the neighbor:" Jesus was insistent that the category of neighbor even includes one's "enemy," and that Christians are commanded to love them as well.
So, how does this relate to this situation? Well, when Jesus wanted to concretize what love actually looks like in real life, he pointed to service: he makes the connection for the lawyer with the parable of the Good Samaritan; and he makes it for the disciples by washing their feet. So, one can make a strong case that for Christians, the paradigmatic way in which they express love for their neighbor is by serving them: giving them food when they need it, clothing when they need it, coffee when they need it, a hotel room, medical attention, a pedicure, etc., etc.
In short, then, this "religious exception" is a complete and total sham--a perversion of a command that stands at the core of the gospel message, and a pitiful attempt to cloak old-fashioned discrimination and persecution in religious dress. I, for one, am not buying it, and it needs to be exposed for exactly what it is. You may not want to serve gays and lesbians, but make no mistake, in acting on that desire, you put yourself unequivocally in opposition to everything Jesus' ministry was about, everything he commanded his followers to do, and everything he died on the cross to free us for. I can think of lots of names for that kind of behavior, but "Christian" is most certainly not one of them.