Licenses to exercise constitutional rights
A commenter on a recent thread suggested that, if the government can require licenses or permits to possess a gun—even licenses that are available to all people who have Second Amendment rights—then the government would have a similar power as to other rights, such as the right to speak. Another commenter suggested that the difference in treatment between guns and speech stems simply from an unprincipled gun exception from "ordinary constitutional law." (I use the term "licenses" and "permits" interchangeably here.)
It turns out, though that licensing requirements have long been sometimes allowed under "ordinary constitutional law"—and sometimes forbidden. As with much of law (and much of life), there is no one clear rule that applies across the board to many very different situations.
Indeed, in one high-profile constitutional law area, licensing requirements are indeed forbidden: Most speakers don't need to get licenses to speak, print, or e-mail.
But even under the First Amendment, the matter is more complicated. Parade organizers may be required to get permits. Ballot signature gatherers may be required to register with the government, and so may fundraisers for charitable causes, though such fundraising is constitutionally protected. Likewise, the Constitution has been interpreted to secure a right to marry, but the government may require that people get a marriage license. (This was so even when a marriage was required in order to legally have sex.) The Takings Clause bars the government from requiring people to leave their land unimproved and thus valueless, but the government may require a building permit before improvements are made.
You don't need a license to ask for assistance of counsel in a criminal case—but the government may require that the counsel get a license, which isn't at all easy to get. Likewise, the government may require that abortion providers (though not abortion recipients) have medical licenses. Many of these requirements are instituted to prevent crime (chiefly fraud) or injury (such as the injury stemming from unsafe construction).
This of course leaves the question of what the right to bear arms is most like: those rights for which government licensing can't be required, or those rights for which it can be. One can debate that question.
But it's a mistake, I think, to assume that the "ordinary constitutional law" rule is that the government may not require a license before you can exercise any constitutional right. Likewise, I doubt that recognizing a licensing requirement for guns will materially affect the well-settled law on when licensing can be required for speech, given that the areas in which licensing of speech may not be required have long endured despite the presence of analogous areas in which licensing of speech may be required.