But if we mean participation in the specific government-fostered institution characterized by marriage licenses, then the answer must be no.
Here's why: if the government-fostered institution were abolished tomorrow, as libertarians favor, no one's rights or freedom would be violated. (Justice Clarence Thomas seems to recognize this in his dissenting opinion.)
We have the inherent right to make contracts but we have no right to anything provided by the state, an inherently coercive organization. That's why the best argument for legal recognition of same-sex marriage is an equal-protection argument, not a liberty argument. It's not so much that we have a right to equal protection; it's that equal protection limits the discretion of government officials -- and that tends to be a good thing. The exception to this equality-but-not-liberty principle would be in those states that both forbid same-sex marriage and refuse to recognize private marriage contracts -- which seems to be all the states affected by the Obergefell ruling. As Ilya Somin writes:
In most states that banned same-sex marriage before today, a same-sex couple could not sign an enforceable marriage contract, even if its content was limited to purely private marital obligations between the two parties.Thus such couples were not only denied equal protection; they were also denied liberty.