Yes, in most instances people wouldn't put up with it. And by not putting up with it, I mean filing litigation against the church. I'm almost certain this will happen at some point.
There are two possible solutions to this 'problem'.
1) Only people acting in the capacity of a state bureaucrat are obliged to issue certificates for those legally permitted to marry.
2) If a pastors is obligated to perform marriages even if they disagree morally with the marriage - and they don't like it they should drop their status as being able to legally marry people and only retain their ability to marry people 'in the eyes of the Lord' and refer the couples that he does marry to the registry office to make it a legal marriage.
A pastor shouldn't have any moral qualms about 'legally marrying but not religiously marrying' someone. He (or she) doesn't have to give it God's blessing, after all...it would just be him carrying out the duties and responsbilities he has implicitly agreed to by signing up to perform state-sanction weddings: like paying taxes or jury duty.