In any case they belong to South America, certainly not to some far away European nation. What is this, the 19th century?
While I'll traditionally root with the continent of half of my heritage, on this matter I'm actually not so sure.
There's a lot of "what's done is done"-ness in human history, especially in terms of settling land.
When you say what you say, you must be refering to geography, because the population on the islands have nothing to do with the South American continent - they are super-British. But when we narrow something down to geographical proximity, we'd have to unravel a whole lot of the world, who owns what and why. Every migration, every conquest, every colony that left behind large bulks of their own culture - such as the entire modern country of Argentina, not to mention nearly every other southern *and* northern American country.
Relating to this issue as a matter of pure geographical proximity, in my opinion, is a little bit arbitrary.
The most pragmatically rational thing to do, would be to grant the islands independence. They'd be neither British NOR Argentinean. However, the population seem to have no such desire, so, British they stay.
(Not to mention, independence for a nation of, what, 2000 people? They'd never make a living)
The Islands, their ownership, their history - and their obvious geographical position, are in themselves all highly contentious issues - by their very own merits. It's not like the Falklands populations have posed some kind of threat - or that the islands are of exceptional strategic significance - so much so as to become a vital geopolitical factor for Argentina or any other such tangible issue. It's just proximity *itself*, the nationality of the population *itself*
French Guyana could theoretically have been in the same situation if Brazil had had any political reason to make a claim and pursue the issue. The reasons would be the same: proximity, European colonialism, a sense of logic "It's basically part of Brazil anyway!"
I know there's also who-owned-what-first, but geopolitics *around the world* is marked by who-owned-what-first, there's territorial claims all over (if not formal claims, there's even more "traditional" claims), but that has always been the way frontiers change - how land is controlled. Whoever exercises physical control is the owner by merit of actively exercising physical control. Argentina did a valiant attempt at exercising physical control over the islands. Britain reasserted physical control. Everybody knows that Argentina is free to try again. Every country in the world is free to try any kind of conquest or liberation or whatever, but they all weigh the cost and the risk and the merit of such a venture. UN and NATO and other organizations make these kinds of moves rare in the modern age, but every now and then it happens, like Ecuador trying to get back some piece of Peru, before UN made them cool it or else sanctions etc.
I guess that could count as somewhat of an unpopular opinion, in particular from someone who speaks Spanish :v