Exactly where is the line drawn of who is and isn't participating in a wedding?
There is not much case law on the subject, but I have read a few law review articles that suggest that courts would examine whether the participation of the actor in the wedding could be construed as an endorsement of the wedding (in violation of his or her religious beliefs).
So the easy case is that of a pastor or minister asked to officiate for a gay couple if he belongs to a church which does not recognize same sex marriages. Even though the wedding certificate is not coming from the church (and the pastor or minister is acting as a quasi-agent of the state), no court would force the pastor or minister to officiate due to the (possible) perception that he or she is endorsing same sex marriages by participating in the wedding ceremony itself. (This perception of endorsement should be objective to an outside observer rather than merely subjective.)
The harder cases are for actors who provide ancillary services to the wedding itself. Take the case of the baker. Pretend a bakery provides different types of wedding cakes.
First, the baker may sell pre-made wedding cake that can be picked up. It makes little sense to argue that a gay couple purchasing a pre-made wedding cake is an endorsement of same sex marriage on behalf the baker. The action is akin to going to Costco to buy a cake.
Second, the baker may sell wedding cakes according to a catalogue. Once again, it makes little sense to claim that ordering "#11 with strawberry and vanilla frosting" is tantamount to an endorsement of same sex marriage.
Third, the baker may sell wedding cakes with personalized messages on it. The type of message probably makes a difference. For example, imagine a wedding cake with the message "Adam & Taylor Forever." It makes little (legal) sense for the baker to agree to make the wedding cake for Adam marrying Taylor Swift, but then object if Adam is marrying Taylor Lautner. The message on the cake does not magically shift between Taylor Swift and Taylor Lautner. On the other hand, if a gay couple wanted a wedding cake with the message "God Bless This Couple Forever," then the baker likely has grounds to object if her or his religious beliefs do not include same sex marriage. Once again, its is about the perception (as viewed by a reasonable third party) that there is an endorsement in violation of the baker's religious tenets (i.e. God cannot bless same sex marriages because homosexuality is sinful).
I wish that I could offer a more precise answer, but there is little case law on the subject, so it will be an evolving standard as the courts establish legal tests for what does and does not count as participation and endorsement.