Religious Freedom doesn't mean your religious beliefs trump the law. As a matter of fact, if there is a conflict, the law usually wins.
For instance, Deuteronomy 22 says you should stone to death brides that can't demonstrate their virginity. The law says nah, you can't do that. That is not a violation of your religious freedom.
Of course that doesn't mean laws that directly impact religion always win. The federal Religious Freedom Restoration Act (RFRA) restricts government from interfering with religious freedom. The key section states “Government shall not substantially burden a person’s exercise of religion even if the burden results from a rule of general applicability."
However even then it's not absolute. There is an exception if two conditions are met.
(1) The burden must be necessary for the "furtherance of a compelling government interest."
(2) The rule must be the least restrictive way in which to further the government interest.
This why clerks who are claiming that issuing marriage licenses to gay couples because of religious objections are wrong. The law both meets a "compelling government interest," equal protection under the law, and using the same procedures in place for currently issuing marriage licenses is about the simplest way to implement it.
My "religion" disagrees with the law is not a valid exception as long as the two conditions are met. This is why clerks which tried to claim a religious exemption from issuing licenses to inter-racial couple got absolutely no where.
This is just another demonstration that conservative evangelical Christianity is an ignorant, bigoted and hateful philosophy that deserves neither adherence or respect.