It's not too hard to see how the concept of a Starfleet Marine Corps might have been dreamed up by several fans over the decades. Star Trek's central exploration organisation, Starfleet, was originally presented with several superficial parallels with the real US Navy, because it was a convenient common reference point for American audiences and writers, steeped as they are in their militaristic culture. And from there, it's not a massively imaginative leap to propose other parallels, never seen on screen, like assuming that there'd also be Starfleet marines to match the US marines (who are officially a branch within the US Navy).
But this is a silly assumption. Those superficial similarities between Starfleet and the US Navy (little more than their rank structure and some basic procedural terminology) were never meant to imply that Starfleet was a military organisation, let alone an exact mirror of an older Earth military organisation. They're conceptually closer to NASA with phasers, not to a military force with warp drives.
But more importantly, there shouldn't even be any Starfleet Marine Corps, whether related to any historical military parallel or not. It would be such a massive misinterpretation of what Starfleet is for, and of what Star Trek is all about. I will now rant in a semi-structured way for several paragraphs to explain why.
First, at the broadest scale, Star Trek is about humanity being better than we used to be. War, violence, and organisations devoted purely to spreading these (a.k.a. militaries) are exactly the sorts of things the show has always presented as the horrible side of modern humans, which have been largely done away with in the future. In that sense, marines don't fit the tone or message of the franchise. If you like the optimism and hope and sense of mutual respect that most Star Trek tries to convey (i.e. if you're a real Trekkie who's actually watched the shows), then you shouldn't want it to be about people whose sole job is to kill.
Second, Starfleet isn't a military organisation. It has a partial, sporadic defence role, compared by Gene Roddenberry to the US Coast Guard's role, but that's secondary to its real goals of exploration and peaceful contact with the galaxy. In canon, there is a Starfleet Diplomatic Corps, as well as numerous science, engineering and medicine divisions. These aren't little side growths of the main thing, they are the main thing, and it's their personnel who populate starships and do the primary work of Starfleet. Similarly, the character of the United Federation of Planets, the state that operates Starfleet, is a peaceful, engaging and supportive one. The Federation wouldn't burden itself with a division (either under Starfleet or autonomous) who do nothing but kill.
It's true that there is a Security division within Starfleet, and they do often go lightly armed (by 23rd/24th century standards). But security officers are not soldiers, they're more like police. Their emphasis is on preventative measures and criminal investigations, with phasers set to stun. (They also have a long history of perishing first on away missions, mostly from things no space marine, nor anyone else without a full name, could survive either.)
We do see Starfleet Security growing larger and more dominant in the period leading up to the Dominion War, but this is presented as a dangerous aberration, not business as normal. DS9 was quite clear that we should fear and reject the militarisation of what ought to be simply a policing service, and reject the militarisation of society in general. The Federation suffers until it re-learns this lesson.
The Dominion War in general was not written and presented as an argument for staying heavily armed and aggressive. It was clearly an anti-war story, about how the Federation's failure to fully understand and engage with the new unknown (the Founders), and its resulting descent into fear of this unknown, could lead to runaway escalation, out of anyone's real control. It was a beautifully crafted mess, entirely believable, and ultimately still reinforcing the core principles of Star Trek idealism. Unfortunately, I notice a lot of fans didn't seem to take away much more from it than "Woo! Lots of ships! Pew pew pew!" This may be related to why some of them want there to be Starfleet marines too.
This is broadly true of any portrayal of war in Star Trek, but the Dominion War is easily the biggest and most compelling example. These wars are not failures of the concept of diplomacy, they are failures to enact diplomacy intelligently and with earnest vigour.
(And while I'm on that, but digressing, I'm baffled by people who think that Section 31 should be accepted as a functional, normal component of the Federation. DS9 explicitly and overtly shows them to be a malfunctioning cancer within the Federation, a hangover from more paranoid and irrational times. Their methods are unacceptable, and their self-selected goals are questionable at best. I can see how they make for exciting story-conflict, shaking up the stability and happiness of main characters. But so does a murderer; doesn't mean we should give them an official salary and a fancy uniform to do their murders in. Section 31 ought to be reviled and rejected by any competent Starfleet Academy graduate, and fans ought to be led to view the concept that way too. Partly, it bugs me that people misunderstand the character of the Federation badly enough to want to keep Section 31 in it, and partly it annoys me that people can't see that this would ruin Section 31's story potential anyway. They only work as interesting antagonists because they're not supposed to be there.)
Some have pointed me towards the MACO organisation, to suggest some sort of support for the concept of Starfleet Marines. But this is a terrible argument, in a few ways. First, the one thing the whole series of Enterprise managed to show quite consistently was how terrible things were in the bad old days before the Federation. 22nd century Earth is better off than we are now, but they still lack a lot of what eventually makes the Federation close to a proper utopia. United Earth was bad at working in space, and they made a lot of dumb mistakes, leading to some pretty big disasters, the Earth-Romulan War being their biggest. That series never got around to showing us exactly how that war unfolded; we just know it shook them up badly for over a century after, and pushed United Earth and its neighbours to enter into the United Federation of Planets. Avoiding another such war was a founding goal of the Federation. The smaller Earth-Xindi conflict that saw the introduction of MACOs to the series was a (somewhat badly written) analogy for the post-9/11 War on Terror, which younger readers may not realise was mostly a load of crap. It was a knee jerk reaction of mindless, unfocused violence, and the MACOs symbolise that exactly (though it's unclear if the writers consciously intended this).
Second, it is now explicit canon that MACO was fully disbanded with the foundation of the Federation, and viewed as unwanted and out of date. The failure of certain former MACOs to integrate into the new Federation Starfleet illustrates just how bad a fit they were for the new Federation era, even in the earliest years. The later centuries of the Federation would be far more alien to the MACO concept. I agree that MACO is a good analogue for a marine-type force, and this only serves to further show how poor a fit such marines are for the Federation.
I have a hypothesis that many of the fans who want there to be a Starfleet Marine Corps are less influenced by the series (except perhaps parts of Enterprise and the shooty pew pew scenes from later seasons of DS9), and are instead re-imagining Starfleet in their own heads, through the combined filter of random bits of Earth history, possibly some unrelated scifi series and movies (many of which feature fully militaristic space navies and some form of space soldiers or marines), and (most crucial of all, I would guess) the many Star Trek games.
Violence is generally more easily gamified than diplomacy or science, and a great many of the more recent Star Trek video and tabletop games have focused on ship combat or infantry combat. I might write up a full survey of this in future, but it shouldn't be too controversial to point out how many Star Trek games have nothing to do with the core activities of the series. And while I think these can still be fun enough games, and I play most of them myself, I think this confuses some fans about what Star Trek is actually trying to be about. I get the feeling that some people are a lot like General Trelane (retired); they like the shiny buttons and ceremonies of militaries, and they think the violence is just good fun. They don't want to acknowledge the harm of it all, and a few even get seriously hostile towards anyone trying to spoil their fun.
I'm especially annoyed at the existence of a marine division in the Trekkie organisation I belong to, STARFLEET International (SFI), which has its STARFLEET Marine Corps (SFMC). STARFLEET is a great fan organisation, I enjoy them, but SFMC is a very strange sort of parasite organisation. Since they have no canon basis for existing, they seem to attract a very separate membership set from the core SFI membership. I've been in multiple SFI chapters since 2012, both badly run and well run ones, and I've seen all sorts of people as members. And my informal assessment so far is that the kind of people who take SFMC seriously and want to participate in it are not interested in the nerdy, geeky science and politics sorts (like me) who participate in regular SFI. It might be some sort of left wing, right wing divide, now that I think of it, but I won't commit to that guess yet. But I digress. There seems to be no logical or canon reason for SFMC to exist, and no love for it from any regular SFI members I've spoken with. Yet it persists, and I think this is more about money than anything else. I suspect SFI would risk losing a huge chunk of membership fees if it cut off SFMC. And presumably SFMC gets a convenient facade of legitimacy and community from wearing the Star Trek and SFI labels. But it bugs the fuck out of me, and I wish they'd leave, or better still, "repent" and join the nicer side of Trek.
Star Trek is idealistic and optimistic and consciously non-violent, by design. The existence of phasers and photon torpedoes raise interesting questions about this ideal future, but they do not invalidate it, and neither do they validate the worst human impulses to band together to kill. The notion of a Starfleet Marine Corps is dumb, because it ignores all of the narrative, aspirational and inspirational conventions of Star Trek. Including marines would fundamentally spoil the stories being told, and they would spoil the social purpose of telling these stories. If you want to get together as a group and kill people, you clearly need to watch a lot more Star Trek.