Donald Sensing has proposed splitting the legal and the spirtual sides of marriage, letting the government offer CICs (Civil Interpersonal Contracts) to any two adults who wanted them, which grant all the legal benefits of marriage without calling it that, while letting the churches (and other religious institutions) issue certificates of marriage, which have no legal bearing while bestowing the name "marriage," with all its spiritual and cultural significance. While there are a couple of things to recommend it, this proposal does have problems. First, the ACLU would be up in arms if certificates of marriage could only be handed out by religious institutions, and I think that any satisfactory compromise would end up with everybody and their brother being able to hand out the things. Second, marriage would become much less common. In Reverend Sensing's view this may be an advantage, as he's not a big fan of people becoming married just for the legal benefits.
The interesting question which he doesn't address is what the Bible says about who can perform marriages, and quite frankly it doesn't say much at all. The Old Testament is full of regulations concerning the contractual obligations of both sides, and the New Testament in particular describes the spiritual aspects. As far as I know, neither one says anything about who performs the wedding, what papers need to be signed, what the ceremony looks like, or what vows are exchanged. From what I've read, I think that the sexual relationship itself makes people married. "Therefore a man leaves his father and his mother and clings to his wife, and they become one flesh." (Gen 2:24) There are a few other verses that seem to support this (Deuteronomy 21:11-13, for example). If this is the case, then any man and woman who have a sexual relationship are married in God's eyes (the Bible clearly doesn't apply this to homosexual relationships), and all the legal, cultural, and ceremonial benefits and obligations are simply to support what is already a spiritual truth.
New Post: More on the subject of the equivalency of sex and marriage above.