Debates about gay marriage take for granted that it is the state that is a proper institution for marrying people. But why should the state marry anyone? The state is a secular institution. It seems appropriate for it to be concerned with regulating contracts, inheritance, retirement and sick benefits, etc., but it could do that without marrying people. With the possible exception of issues surrounding parenting of minor children or cases involving adults with diminished mental capacity, it is not obvious why the state should impose ANY restrictions on who can form a civil union or create some other secular contractual relationship.
It is apparent that much of the objection to gay marriage is because it is seen as contrary to the teaching of various religions. If marriage becomes the concern of non-governmental organizations such as churches, each of these organizations could impose whatever rules it wished to impose on couples who wished to marry within that organization. Secular individuals who have no interest in being part of one of these organizations need not affiliate with them. There would be no obvious reason for them to employ ministers to provide a dollop of Christian spirituality in an otherwise thoroughly secular (or pagan) event.
Republicans like to privatize stuff. Here's a chance for them to show off their enthusiasm for privatization in a way that will not diminish the delivery of public goods like education, clean air and water, public safety, etc.