1. The Cosby Extortion Case. As Eugene notes, someone with an underlying legal claim may threaten to expose it to reach a reasonable settlement. Yet if the amount sought is so substantially out of line with the injury and the threat to embarrass is a big part of the threat, then a criminal charge of extortion can be established.
The chief example here is the Cosby case, where Autumn Jackson, who may have been Cosby's daughter, threatened exposure unless he paid her $40 million. Despite having some possible claim for support as a child, she was convicted because of the excessiveness of her claims and the threats of exposure. Eugene notes that the 2d Circuit Court of Appeals (initially) reversed the conviction of Autumn Jackson. But on rehearing, the 2d Circuit reversed itself and reinstated Jackson's conviction because the error in her jury instructions was harmless. Thus, she was found guilty and her conviction was affirmed.
Here are some excerpts on the law from the 2d Circuit's Jackson/Cosby opinions:
[From 180 F.3d at 81:] The evidence at trial was plainly sufficient to support verdicts of guilty had the jury been properly instructed. Even if Jackson were Cosby's child, a rational jury could find that her demand, given her age (22) and the amount ($ 40 million), did not reflect a plausible claim for support. The evidence supported an inference that Jackson had no right to demand money from Cosby pursuant to a contract or promise and no right to insist that she be included in his will. The jury thus could have found that her threat to disclose was the only leverage she had to extract money from him; that if she sold her story to The Globe, she would lose that leverage; and that if Cosby had capitulated and paid her in order to prevent disclosure, there was no logical guarantee that there would not be a similar threat and demand in the future. Thus, had the jury been instructed that the "with intent to extort" element meant that defendants could be found guilty of violating § 875(d) only if Jackson's threat to disclose was issued in connection with a claim for money to which she was not entitled or which had no nexus to a plausible claim of right, the jury could permissibly have returned verdicts of guilty on that count.
[From 196 F.3d at 387:] We reasoned, however, that not all threats to reputation accompanied by demands of money are inherently wrongful, see id. at 67, and we inferred that Congress meant § 875(d) to criminalize only such threats as are wrongful, see id. at 67-70. We concluded that the court's instruction "erroneously allowed the jury to find defendants guilty of violating that section on the premise that any and every threat to reputation in order to obtain money is inherently wrongful," id. at 71-72, and that the court should instead have informed the jury of the wrongfulness element by instructing that § 875(d) prohibits obtaining money or a thing of value from another by use of threats to reputation only if the defendant has no plausible claim of right to the money demanded or if there is no nexus between the threat and the defendant's claim, see 180 F.3d at 71.
Further, in State v. Harrington, 260 A.2d 692 (Vt. 1969), a lawyer for the wife in a fault-based divorce action was convicted of attempting to extort a settlement by threatening to expose a relevant part of the case, the husband's extramarital affair. The lawyer was also disbarred. The threat to go to the newspapers was explicit and the amount seems to have been excessive.
2. Is the threatening letter posted on the BlogActive website a crime? First, one must look to federal or state statutes to determine their scope. Some blackmail or extortion statutes, such as the federal Hobbs Act, punish only threats seeking property. And the US Supreme Court took a narrow view of property in Scheidler v. NOW II (2003). [DISCLOSURE: I consulted and worked on the brief for NOW in Scheidler v. NOW I, which NOW won in the Supreme Court, and I mooted the NOW attorneys in NOW II, which NOW lost. The case is currently before the Supreme Court yet a third time in Scheidler v. NOW III.] Some state statutes punish both obtaining property and compelling action under the same extortion statute. Other state statutes divide the traditional crime of extortion into one covering property and another covering compelling action (often called criminal coercion).
So this threatening letter would probably not be extortion under the Hobbs Act because it seeks to compel action (voting), not to obtain property. There may be other federal statutes it might violate; I don't know.
But the posted letter may well be extortion or criminal coercion if committed in many, perhaps most, states. Whether it is a crime turns primarily on the closeness of the nexus between the threat (exposure) and the action sought (voting against Alito).
Typically, in order to avoid a violation of an extortion statute, the threat must be very closely linked to the underlying claim. Clearly asking for much more than you are owed (such as in the Jackson/Cosby case) under a threat of exposing embarrassing behavior was enough to lead to Autumn Jackson's conviction.
Similarly, some statutes recognize a defense of seeking only restitution or seeking only to right a wrong in circumstances related to the underlying claim. It would seem to be this defense that the threatener at BlogActive would want to try to claim. It appears that in Mike Rogers's mind, a male Senator having sex with another man in the bathroom of Union Station is directly related to that Senator's vote confirming Justice Alito. Although there isn't enough case law here to give a definitive answer, I suspect that a court would not find the nexus between voting for Alito and gay sex in a public bathroom to be close enough to allow Rogers to use that defense. The nexus in the Jackson/Cosby case and in the Harrington case would seem to have been closer, and both those defendants nonetheless had their convictions affirmed.
The kinds of threats that would seem to be covered by the defense (and thus not extortion) are such threats as:
Pay me back the money you stole or I'll go to the police.
Pay your employer back the money you stole from the company or I'll tell your employer of your theft.
Stop stealing bicycles or I'll report you to the police.
Stop having sex with men in public bathrooms or I'll expose that you are having sex with men in public bathrooms.
Note that the last two examples involve compelling action, but the nexus between the threat of exposure and the action sought is very close.
UPDATE: As to the jurisdictional location of the threat, I consciously avoided that issue. Two commenters below quote the DC blackmail statute, which is not unusual except that it still uses the term "blackmail," which is less common today than using "extortion," "theft," or "coercion." It definitely covers compelling action.
Washington D.C. Criminal Code section 22-3252 provides:
Note that the DC statute does not have an explicit claim of right defense. That doesn't mean there is isn't such a defense (there must be to make sense of the scope of the law), though I seem to recall that one state struck down its extortion statute because of an inadequately broad claim of right defense (on a facial overbreadth challenge).
(a) A person commits the offense of blackmail, if, with intent to obtain property of another or to cause another to do or refrain from doing any act, that person threatens:
(1) To accuse any person of a crime;
(2) To expose a secret or publicize an asserted fact, whether true or false, tending to subject any person to hatred, contempt, or ridicule; or
(3) To impair the reputation of any person, including a deceased person.
(b) Any person convicted of blackmail shall be fined not more than $1,000 or imprisoned for not more than 5 years, or both.
(1981 Ed., § 22-3852; Dec. 1, 1982, D.C. Law 4-164, § 152, 29 DCR 3976.) Washington D.C. Criminal Code section 22-3252
Some have raised keeping the victim's identity secret. In the US, usually no effort is made to protect the identity of blackmail victims (unlike some rape victims). Such efforts would probably be futile for someone as prominent as a US Senator.