I was probably being too general in my earlier post. When it comes to the FFCC, States generally have an obligation to enforce money judgments awarded in other states, and abide by other judicial decrees issued by other states (such as divorce decrees). But states can generally set their own rules as to what people can do in the state -- they can control who can drive, who can practice law, who can practice medicine, who can engage in other businesses, who can carry a gun, who (if anyone) can buy alcohol, and so on. That's part of the state's sovereign power.
So as a result, states need not honor all sorts of licenses issued by other states. In practice, there's a good deal of "comity" -- voluntary accommodation of other states' decisions. I believe that this is why states honor out-of-state driver's licenses; they don't have a constitutional obligation to do this, but they do, for good pragmatic reasons. But as a general matter, one state may not export its gun carry licensing policy to other states unless the other states acquiesce. If it were otherwise, then no state could prohibit, say, gambling, prostitution, and such; after all, California would have to respect Nevadan licenses to gamble and to be a prostitute.