No, there's no 'hate speech' exception to the First Amendment
I keep hearing about a supposed "hate speech" exception to the First Amendment, or statements such as, "This isn't free speech, it's hate speech," or "When does free speech stop and hate speech begin?" But there is no hate speech exception to the First Amendment. Hateful ideas (whatever exactly that might mean) are just as protected under the First Amendment as other ideas. One is as free to condemn Islam—or Muslims, or Jews, or blacks, or whites, or illegal aliens, or native-born citizens—as one is to condemn capitalism or Socialism or Democrats or Republicans.
To be sure, there are some kinds of speech that are unprotected by the First Amendment. But those narrow exceptions have nothing to do with "hate speech" in any conventionally used sense of the term. For instance, there is an exception for "fighting words"—face-to-face personal insults addressed to a specific person, of the sort that are likely to start an immediate fight. But this exception isn't limited to racial or religious insults, nor does it cover all racially or religiously offensive statements. Indeed, when the City of St. Paul tried to specifically punish bigoted fighting words, the Supreme Court held that this selective prohibition was unconstitutional (R.A.V. v. City of St. Paul (1992)), even though a broad ban on all fighting words would indeed be permissible. (And, notwithstanding CNN anchor Chris Cuomo's Tweet that "hate speech is excluded from protection," and his later claims that by "hate speech" he means "fighting words," the fighting words exception is not generally labeled a "hate speech" exception, and isn't coextensive with any established definition of "hate speech" that I know of.)
The same is true of the other narrow exceptions, such as for true threats of illegal conduct or incitement intended to and likely to produce imminent illegal conduct (i.e., illegal conduct in the next few hours or maybe days, as opposed to some illegal conduct some time in the future). Indeed, threatening to kill someone because he's black (or white), or intentionally inciting someone to a likely and immediate attack on someone because he's Muslim (or Christian or Jewish), can be made a crime. But this isn't because it's "hate speech"; it's because it's illegal to make true threats and incite imminent crimes against anyone and for any reason, for instance because they are police officers or capitalists or just someone who is sleeping with the speaker's ex-girlfriend.
The Supreme Court did, in Beauharnais v. Illinois (1952), uphold a "group libel" law that outlawed statements that expose racial or religious groups to contempt or hatred, unless the speaker could show that the statements were true, and were said with "good motives" and for "justifiable ends." But this too was treated by the Court as just a special case of a broader First Amendment exception—the one for libel generally. And Beauharnais is widely understood to no longer be good law, given the Court's restrictions on the libel exception. See New York Times Co. v. Sullivan (1964) (rejecting the view that libel is categorically unprotected, and holding that the libel exception requires a showing that the libelous accusations be "of and concerning" a particular person); Garrison v. Louisiana (1964) (generally rejecting the view that a defense of truth can be limited to speech that is said for "good motives" and for "justifiable ends"); Philadelphia Newspapers, Inc. v. Hepps (1986) (generally rejecting the view that the burden of proving truth can be placed on the defendant); R.A.V. v. City of St. Paul (1992) (holding that singling bigoted speech is unconstitutional, even when that speech fits within a First Amendment exception); Nuxoll ex rel. Nuxoll v. Indian Prairie Sch. Dist. # 204, 523 F.3d 668, 672 (7th Cir. 2008) (concluding that Beauharnais is no longer good law); Dworkin v. Hustler Magazine Inc., 867 F.2d 1188, 1200 (9th Cir. 1989) (likewise); Am. Booksellers Ass'n, Inc. v. Hudnut, 771 F.2d 323, 331 n.3 (7th Cir. 1985) (likewise); Collin v. Smith, 578 F.2d 1197, 1205 (7th Cir. 1978) (likewise); Tollett v. United States, 485 F.2d 1087, 1094 n.14 (8th Cir. 1973) (likewise); Erwin Chemerinsky, Constitutional Law: Principles and Policies 1043-45 (4th ed. 2011); Laurence Tribe, Constitutional Law, §12-17, at 926; Toni M. Massaro, Equality and Freedom of Expression: The Hate Speech Dilemma, 32 Wm. & Mary L. Rev. 211, 219 (1991); Robert C. Post, Cultural Heterogeneity and Law: Pornography, Blasphemy, and the First Amendment, 76 Calif. L. Rev. 297, 330-31 (1988).
Finally, "hostile environment harassment law" has sometimes been read as applying civil liability—or administrative discipline by universities—to allegedly bigoted speech in workplaces, universities, and places of public accommodation. There is a hot debate on whether those restrictions are indeed constitutional; they have generally been held unconstitutional when applied to universities, but decisions are mixed as to civil liability based on speech that creates hostile environments in workplaces (see the pages linked to at this site for more information on the subject). But even when those restrictions have been upheld, they have been justified precisely on the rationale that they do not criminalize speech (or otherwise punish it) in society at large, but only apply to particular contexts, such as workplaces. None of them represent a "hate speech" exception, nor have they been defined in terms of "hate speech."
For this very reason, "hate speech" also doesn't have any fixed legal meaning under U.S. law. U.S. law has just never had occasion to define "hate speech"—any more than it has had occasion to define rudeness, evil ideas, unpatriotic speech, or any other kind of speech that people might condemn but that does not constitute a legally relevant category.
Of course, one can certainly argue that First Amendment law should be changed to allow bans on hate speech (whether bigoted speech, blasphemy, blasphemy to which foreigners may respond with attacks on Americans or blasphemy or flag burning or anything else). Perhaps some statements of the "This isn't free speech, it's hate speech" variety are deliberate attempts to call for such an exception, though my sense is that they are usually (incorrect) claims that the exception already exists.
I think no such exception should be recognized, but of course, like all questions about what the law ought to be, this is a matter that can be debated. Indeed, people have a First Amendment right to call for speech restrictions, just as they have a First Amendment right to call for gun bans or bans on Islam or government-imposed race discrimination or anything else that current constitutional law forbids. Constitutional law is no more set in stone than any other law.
But those who want to make such arguments should acknowledge that they are calling for a change in First Amendment law, and should explain just what that change would be, so people can thoughtfully evaluate it. Calls for a new First Amendment exception for "hate speech" shouldn't just rely on the undefined term "hate speech"—they should explain just what viewpoints the government would be allowed to suppress, what viewpoints would remain protected, and how judges, juries, and prosecutors are supposed to distinguish the two. Saying "this isn't free speech, it's hate speech" doesn't, I think, suffice.