There are some social conservatives who plan to propose a constitutional amendment to protect this type of "religious liberty":
“A religious organization, religious association, religious society or any person acting in a role connected with such organization, association or society and shall not be required to solemnize, officiate in, or recognize any particular marriage or religious rite of marriage in violation of its constitutional right of conscience or its free exercise of religion.”Again, this is a great concept in theory (ignoring the fact that the First Amendment makes it wholly unnecessary), but the ways it would be used undermine its innocent appearance. Before and after Loving v Virginia, the Supreme Court case that overturned state bans on interracial marriage, the same argument was made. Religious reasons were given to ban interracial marriage, and religious liberty was claimed after the bans were overturned so that churches wouldn't have to perform interracial marriage. It didn't work then, and it won't work now.
People have their marriages performed in their church. If they don't go to a church, they have it performed in a public place by a public official. If a same-sex couple requests that a church perform their marriage, they were almost certainly already parishioners of that church, in which case the church is already accepting of their relationship and union. No one is going to go into a church they have no relationship with and ask to have their wedding performed by someone who doesn't approve of it. People are asking for this type of religious "liberty" to protect them from something that is entirely fictional.
But this isn't only about marriages. Religious liberty is self-evidently important, since it also includes the broader freedom of conscience. However, as with any liberty, yours ends where it infringes on another. Religious liberty doesn't allow you to commit crimes by claiming your religion allows it, nor should it allow you to exclude certain people (not actions) from your association. The Boy Scouts (before this year) are a perfect example. They claimed the right to exclude gays from their organization because of their religious opposition to homosexuality. This claim undermines itself. Of course homosexual activity can be banned by a private organization, especially since they also ban heterosexual activity within the organization. However, to ban people with a homosexual or bisexual orientation crosses the line. They're not being prevented from practicing their religion or holding any religious or non-religious opinion by having to simply allow certain types of people in their organization.
If your religion opposes eating lobster (which Biblical religions technically do), you can ban eating lobster at your church, but you can't ban people from your church who happen to like eating lobster.