Carry permits have been unsuccessfully argued time and time again under the reciprocity clause. This same-ex marriage ruling doesn't alter the arguments.
The Full Faith and Credit Clause—Article IV, Section 1, of the U.S. Constitution—provides that the various states must recognize legislative acts, public records, and judicial decisions of the other states within the United States. It states that "Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State." The statute that implements the clause, 28 U.S.C.A. § 1738, further specifies that "a state's preclusion rules should control matters originally litigated in that state." The Full Faith and Credit Clause ensures that judicial decisions rendered by the courts in one state are recognized and honored in every other state. It also prevents parties from moving to another state to escape enforcement of a judgment or to relitigate a controversy already decided elsewhere, a practice known as forum shopping.
In drafting the Full Faith and Credit Clause, the Framers of the Constitution were motivated by a desire to unify their new country while preserving the autonomy of the states. To that end, they sought to guarantee that judgments rendered by the courts of one state would not be ignored by the courts of other states. The Supreme Court reiterated the Framers' intent when it held that the Full Faith and Credit Clause precluded any further litigation of a question previously decided by an Illinois court in Milwaukee County v. M. E. White Co., 296 U.S. 268, 56 S. Ct. 229, 80 L. Ed. 220 (1935). The Court held that by including the clause in the Constitution, the Framers intended to make the states "integral parts of a single nation throughout which a remedy upon a just obligation might be demanded as of right, irrespective of the state of its origin."
The Full Faith and Credit Clause has been applied to many matters, such as child custody. Back in the 70s and 80s other states did not accord child custody determinations with full faith and credit enforcement. That allowed (and even encouraged) one parent kidnapping a child and escaping to another state in order to petition for custody. The Full Faith and Credit Clause put an end to that. It also gave a new measure of protection to victims who moved to a different state after obtaining a protective order in one state regarding domestic abuse.
Traditionally, every state honored a marriage legally contracted in any other state, and as a result marriage has fallen de facto under common law under the the Full Faith and Credit Clause. That's why when Hawaii became the first state to recognize same-sex marriage, the writing was on the wall for other states to at the very least honor marriages that took place in Hawaii. It didn't mean the other states had to start allowing same-sex marriages in their state, but they had to recognize one from Hawaii. Part of the reason this got to the Supreme Court is because several states didn't wanna.
But the Full Faith and Credit has very little to do with the issuing of state licenses. It has to do with legal decisions. If you are a licensed pest control operator in one state, you're still gonna have to get a license to kill bugs in another state. Same with being licensed to practice law or conceal and carry a gun, and even to drive. It's important to keep in mind that that the fact that a person is licensed to do something in State A (carry a gun, practice law, drive) doesn't necessarily mean that State B must honor that license when that person wants to do the same thing in State B. Full Faith and Credit does not apply to that, though many states may choose to honor out-of-state licenses as a matter of policy (as, for instance, with driver’s licenses).
