So, this happened:
This was once relatively common, but not in the last half-millennium and not on Staten Island, so it's newsworthy at this point.
The defendant, Richard Luthmann, made this demand in a reply brief he filed in Richmond County, New York, on July 24. (Credit to SILive.com, Volokh Conspiracy, and Above the Law for earlier reports.) Luthmann is a lawyer there, and while I haven't checked out the other pleadings in this dispute because who cares, based on Luthmann's brief this is some sort of fallout from a previous lawsuit against some of Luthmann's clients. (I'm tempted to read the opposition to which this replies, because Luthmann describes it as "a glorified comic book piled on top of pure and adulterated extortion wrapped in a transparent abuse of legal process," but it probably isn't really a comic book.) That gives you some idea of the rhetoric involved, but here we are interested in the second half of the reply, beginning with paragraph 26, supra.
As it happens, I am in the process of researching trial by combat for a chapter in my next book, and I know that for most of you it's news that there was a first book but that's fine. Luthmann's brief goes into some detail about the history of the "wager of battle," as this practice was also called. His material appears to be largely derived from Wikipedia, but far be it from me to criticize that practice—as long as you check the sources, and there are some decent sources here.
The very short version is that while people have been resolving disputes with violence for as long as there have been people, or at least since people invented beer, here we are talking about a legally sanctioned practice of resolving a dispute by fighting. That is, not just a fight or a duel or two guys fighting in single combat before a battle, but a recognized legal right to demand the other party fight it out. And there was such a thing, of course, apparently most common among Germanic cultures.
Most sources say that the Normans (who had fairly recently been Vikings) brought it to England after the Conquest, although I seriously doubt it hadn't been used there before that. Anyway, in the following centuries it was clearly a formalized practice, and in fact this article, which is very interesting until it veers off into the vast and marshy snoreland that is law-and-economics theory, says that until 1179 "it was England's primary trial procedure for deciding land ownership disputes." It's not clear how often an actual fight took place, but they had the whole process of selecting champions and so on, just like in Game of Thrones (of which, not coincidentally, Luthmann is said to be a huge fan).
The use of this method dwindled over the years—the jury system being usually less fatal—but it remained available until 1819, when Parliament finally abolished it after a rather embarrassing incident the year before in the case of Ashford v. Thornton. There have been a couple of more recent attempts to invoke this right in the UK, the more amusing one being the 2002 case in which a guy who'd gotten a traffic ticket demanded that the Driver and Vehicle Licensing Agency name a champion for trial by battle. (Sadly, a court ruled against him.)
The argument Luthmann is throwing out there, almost certainly tongue-in-cheek, is that in 1776, Parliament had not yet outlawed trial by combat, and so it was still part of the common law at the time the U.S. became independent. Congress has never outlawed it, and the Ninth Amendment (so the argument goes) preserves the right. I don't think any U.S. court has ever ruled on this argument, but it seems very unlikely to work.
Even if it was technically part of the common law 2.5 centuries ago, history and practice since then has not been too favorable, at least by analogy to the once-widespread practice of dueling. That was once not uncommon in certain circles, to the point that some state legislatures did feel the need to take action. (This is why new lawyers in Kentucky have to swear they've never fought a duel, as I wrote about here.) But it was never a judicially sanctioned practice, as far as I know, just socially tolerated. And since it was not infrequently considered murder, that would not be a great precedent for you to invoke.
At the very least, I would urge you not to try this until after you read my book, assuming I get it finished and published and can persuade somebody to buy it for you for Christmas. Either that'll provide useful guidance, or it'll take me long enough that you'll have cooled off by then.