After the Supreme Court ruled in Obergefell v. Hodges that same-sex couples have the same right to marry as opposite-sex couples, Davis stopped issuing any marriage licenses at all. Her actions are clearly discriminatory. Were the Supreme Court to reverse its decision in Obergefell, she would presumably resume her lawful duty to issue marriage licenses to all couples who are qualified under the law. Her decision to stop issuing licenses is analogous to those of certain southern states and counties who unconstitutionally closed the public schools rather than comply with Brown v. Board of Education. The Constitution is not blind to the reasons for government action or inaction.
The first provision of the Bill of Rights is the Establishment Clause, and the second is the Free Exercise Clause. The First Amendment states:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof.Were Davis acting as a private party -- as a wedding photographer or baker or florist -- neither the Establishment Clause not the Equal Protection Clause would apply. But Davis is a government employee, and her official actions are attributable to the government -- they are state action. While she has a personal right to exercise her religion, as a government official she may not to any extent establish religion or violate other people's rights to equal protection of the law. Yet that is precisely what she wishes to do. She claims the right, based upon her personal religious beliefs, to take official action in accord with her religion in discriminating against same-sex couples who seek to marry. Not only does this violate those couples' rights to equal protection; it violates their right to freedom from government establishment of religion.
Government officials are free to express their religious beliefs in their personal lives as well as in speeches and editorials. The government may even conduct ceremonial prayers in settings that are not coercive. But freedom of religion does not give a government official the right to refuse to perform her duty under the law. Quite the contrary. Under the Establishment Clause the government may not even coerce individuals into participating in religious exercise; it certainly may not use religion as a reason to violate the law and deprive people of their rights.