There are three related issues when it comes to national CCW, or national CCW reciprocity (two very different ideas).
The first is whether or not the 2nd does or should fall under the umbrella of the 14th. Obviously, current doctrine is that it doesn't (the SCOTUS has not seen fit to "selectively incorporate" the 2nd), but there's an easy case to be made that it should. While the states' rights argument is conceptually valid, until the 14th is repealed, it simply doesn't have any legal standing (we can argue until the cows come home about whether or not the 14th is valid, but unless it's removed from the Constitution, it's the law of the land). Incorporation of the 2nd as an individual right under the 14th is, at least in my mind, the "holy grail" of the RKBA fight in this country.
The second is whether CCW itself is a second amendment issue, and, much though I wish it were otherwise, I don't necessarily think that it is (I'm open to arguments on it, though). The right to carry a firearm at all times is guaranteed under the 2nd, but the right to carry it concealed isn't. In my opinion, no state has the right to tell me I can't carry a gun, but states do have the right to tell me it has to be in plain view* (or, alternatively, that it has to be concealed).
There's certainly a point of view that any law which restricts how I carry a gun infringes my right to bear arms, but I don't hold with it. For example, I'm fine with laws that prohibit me from carrying a gun by aiming it at passers-by while I walk down the street.
Third, of course, is the full faith & credit clause of the Constitution, and whether this means that every state has a responsibility to recognize the licenses granted by others. This one is a resounding "no." I know it doesn't apply to lawyers, doctors, or nurses, so I imagine it doesn't apply to teachers, veterinarians, barbers, or anyone else professionally licensed by a state. Insofar as CCW is a licensed privilege (see point #2 above), it does not fall under the full faith & credit clause. The DL issue is one that the states worked out between themselves; each state has chosen to recognize the DL of every other state (this "Real ID" thing will probably have some sort of impact on this, but we've yet to see what).
*Yes, I realize this opens the door to onerous and/or unmeetable requirements for "plain view" and/or "concealed," but those would be their own violations of the 2nd. The principle, IMHO, is sound.