The first point to make here, and the most important one, is that no one has the authority to reject or defy a constitutional decision from the Supreme Court. That is how our system of government works and has worked for the whole of our history. For better or for worse, the Supreme Court has the last word on the meaning of the Constitution.
The court's power to enforce its decisions is, of course, limited. Their practical efficacy may depend on how willing the other branches of the federal government are to support the court. In 1832, the Supreme Court decided Worcester v. Georgia, which recognized the rights of Native American tribes. President Andrew Jackson was supposed to have said "John Marshall has made his decision; now let him enforce it!" Jackson was not willing to enforce the decision, and it ended up being of little practical effect.
By contrast, in 1957, President Eisenhower sent the 101st Airborne to Little Rock, Arkansas, to ensure that African-American children could attend the schools to which the Constitution and Brown v. Board of Education entitled them.
Will Obergefell meet explicit defiance? Justice Scalia's dissent gestured in this direction, warning that the court had moved "one step closer to being reminded of our impotence." But it seems unlikely.
The court's desegregation decisions met massive resistance because they directly affected the objecting whites. Its interracial marriage decision, Loving v. Virginia, did not -- presumably in part because there was there was no similar tangible consequence for its opponents.
Obergefell is of course more like Loving than Brown. It is hard to imagine state governors directing county clerks to withhold same-sex marriage licenses, or calling out the National Guard to protect them from the contempt citations that would follow. (And if it came to that, I expect the Obama administration would back the Court.)
So outright defiance is neither legally justifiable nor plausible in practice. But there is a real question about what latitude states have to accept the Supreme Court's ruling while accommodating the sincere religious beliefs of their employees. Yesterday, Texas Attorney General Ken Paxton issued an opinion on this subject, concluding that some degree of accommodation was permissible. This is likely true, though not for quite the reasons Paxton
Paxton invoked three sources
of protection for government employees' religious liberty: the First Amendment to the Constitution, and state and federal Religious Freedom Restoration Acts (RFRAs). The Obergefell decision, he claimed, "stops at the door of the First Amendment and our laws protecting religious liberty."
If the claim is that those sources of law can limit the force of the decision, this is not true. If there were a conflict between Obergefell and the First Amendment, Obergefell would prevail. It interprets a constitutional amendment (the Fourteenth) which was enacted after the First and which could have repealed it, just as the Eighteenth Amendment enacted Prohibition and the Twenty-First repealed it.
But there is no conflict: According to the Supreme Court, the First Amendment offers no protection against generally applicable laws that regulate conduct rather than belief. The requirement to allow same-sex couples to marry is general and pertains to conduct; it cannot be resisted on First Amendment grounds.
The Federal Religious Freedom Restoration Act has no application at all; as construed by the Supreme Court, it is essentially a guide to interpreting federal statutes. Certainly it cannot limit the force of a Supreme Court constitutional decision.
That leaves Texas state law. State law cannot excuse a state from complying with a Supreme Court decision either, but it can affect the way in which a state complies. Texas has an obligation to allow same-sex couples to marry. It may be able to meet that obligation in a way that accommodates the religious beliefs of state employees.
If, for instance, there are two clerks in an office, only one of whom objects to same-sex marriage, it would be consistent with Obergefell to allow the non-objector to issue same-sex licenses -- as long as this practice neither stigmatized same-sex couples nor imposed significant burdens on them.
And in this sense, Paxton is right. The state of Texas must issue same-sex marriage licenses, but in appropriate circumstances, individual clerks may be excused.