And, as an aside, is there such a thing as American monarchists today?
edit: I'm not really asking about Loyalists, rather people who would install a new ''American King'' entirely separate from the British monarchy
I don't know that there has ever been a serious movement, but it's certainly been proposed by people here and there. In 1782 Colonel Lewis Nicola wrote a letter to George Washington suggesting that Washington become the King of the United States in a constitutional monarchy. His reasoning was that a monarchy is something to bind the nation together in times of trouble, while the United States was squabbling and having trouble pooling resources together.
He compares the trouble that Venice, Genoa, and Holland have had. That Holland, despite controlling half the world's commerce, could not stand up to the British and needed to get help from a neighboring monarchy, and the similarities between the US government and these republics would cause similar problems.
In contrast to this scene let us consider the principal monarchies of Europe, they have suffered great internal commotions, have worried each other, have had periods of vigour & weakness, yet they still subsist & shine with lustre. It must not be concluded from this that I am a partisan for absolute monarchy, very far from it, I am sensible of all its defects, the only conclusion I would draw from the comparison is, that the energy of the latter is more beneficial to the existence of a nation than the wisdom of the former. A monarch may often be governed by wise & moderate councels, but it is hardly possible for large bodies to plan or execute vigorous ones.
...
Some people have so connected the ideas of tyranny and monarchy as to find it very difficult to separate them, it may therefore be requisite to give the head of such a constitution as I propose, some title apparently more moderate, but if all other things were once adjusted I believe strong argument might be produced for admitting the title of king, which I conceive would be attended with some material advantages.
link including images and text of letter
Washington was not amused by the idea and rejected it outright.
I am much at a loss to conceive what part of my conduct could have given encouragement to an address which to me seems big with the greatest mischiefs that can befall my Country.
Full text of response: http://www.freedomshrine.com/historic-documents/washington-letter.php
To any historians: is there any validity to the claim that Nathaniel Gorham (head of the Continental Congress) wrote to the King of Prussia asking him to rule over the United States? The only sources I can find do not include citations, or if they do, it's to a 105-year-old book which I can't dig in to.
During the Constitutional Convention, Alexander Hamilton gave a speech in which he advocated for a system of 'elected monarchy.' While there is no transcription of the speech, we have notes from Hamilton himself and from James Madison. Essentially, he argued that the president should serve for life, but could be removed (impeached) for bad behavior. While he did not propose a true monarchy (a new president would be elected on the office-holder's death, rather than passing down from father to son), this speech (combined with Hamilton's admitted fondness for the British system as a whole) led to his political opponents branding him as a monarchist. This label would follow him until his death in 1804, and was a frequent criticism used by the Democratic-Republicans to disparage both Hamilton and the Federalist party as a whole.
John Adams was also frequently attacked as a monarchist, during both his Vice presidential and Presidential terms. These attacks largely began with the strong opinions he expressed during the Senate debate on titles for the President. His desire for a grand title for the president 'His highness, the President of the United States of America, and Protector of the Rights of the Same,' was also seen as an un-Republican attempt to transform the office into a more Monarchical position.
While Adams and Hamilton were both Federalists, monarchical tendencies were by no means exclusive to that party. Aaron Burr, former Vice President (and famed shooter of Alexander Hamilton) conspired to create an 'American Empire' in the west and install himself as a hereditary monarch.
Finally, and less seriously, the threat of monarchy was occasionally (and somewhat passive-aggressively) used by northern states during the Constitutional Convention when they felt that they were being pushed around by the Southern States. Essentially the argument went 'we don't really need to be a part of this nation, I'm sure that there is some European price who will be happy to come be our king.' Obviously that didn't happen, and it was more of a bargaining tool than an actual political proposal. Interestingly, this threat would re-occur throughout the post-revolutionary, generally as a response to perceived overreach by Democratic-Republican presidents.
While the threat of monarchism was frequently brought up throughout the Revolutionary period, no organized political party advocated for a European-style monarchy on a national scale.
Sources:
Alexander Hamilton by Ron Chernow
James Madson, life Reconsidered by Lynn Cheney
John Adams by David McCullough (also a great HBO miniseries)
Jefferson's Message to Congress Regarding the Burr Conspiracy
This is a adaptation of an older answer of mine, but there was a debate about whether our democracy in appearance was going to be more monarchist. This article from the Boston Globe's "Ideas" section doesn't address your core question and isn't about a monarchist party per se, but rather the debate about titles in the early American Republic, which I think shows how differently democracy in America was being imagined in our earliest years. It focuses on Washington's taking power, and specifically the debates about which honorific to use with the Presidency: "His Elective Highness, President Barack Obama" (the article is about Washington, not contemporary politics). Needless to say, the lack of pageantry associated with the presidency disappointed some people in America.
It's short and well written, but to give people a sense of it, here's a preview:
The Senate went on record to recommend the title of “His Highness the President of the United States of America, and Protector of their Liberties,” which senators saw as in keeping with global title conventions. As the debate unfolded in three weeks of closed-door wrangling, the chamber also considered some curious hybrids, like “Elective Highness” and “Elective Majesty”—whose strangeness to present-day ears is a testament to the novelty of the American experiment at the time.
If you want to widen your definition just a little bit, there has been a very dedicated American monarchy, who have at various times (and, in some ways, up to the present day) tried to assert themselves over an oppressive democratic hegemony.
I'm referring to the Kingdom of Hawaii, which remains symbolically powerful in the movement for Hawaiian sovereignty. I think Jonathan Kamakawiwo'ole Osorio has written the most about the line of Kamehameha and King David Kalakaua, who's usually credited with reviving the sport of surfing and the art of hula in the 1800s.
I don't think even the fringier of Hawaiian sovereignty supporters want to see a Hawaiian monarch ruling the rest of the U.S., but putting a traditional king back on a throne in Hawaii is something that gets advocated, or at least mentioned, every so often; one of the movement's key historical grievances is the way the traditional monarchy was overthrown with U.S. support in January 1893.
Slightly less serious, but more wonderful, was the reign of notable San Francisco eccentric Joshua Abraham Norton as Norton I, Emperor of America and Protector of Mexico.
He could be dismissed as a crank, except he was listed in the 1870 census has having the occupation "Emperor," issued currency that was accepted by local merchants, and, after passing on, had a funeral with a cortege over 2 miles long. Mark Twain wrote the imperial dog's epitaph, though had left San Francisco at the time of his majesty's death.
The Virtual Museum of San Francisco has more on Norton I's reign, including the interesting idea that the Bay Bridge was his idea.
Follow-up questions: I have heard that there was such a movement specifically among the land-owning elite in the South leading up to the civil war. Is that true?