Thank you  "If a country's position on the matter receives significant coverage in high-quality secondary sourcing (mainstream media and scholarly works), it is worth inclusion". It's basically the standard for WP:WEIGHT I've been proposing for a while now (except that I offered to adhere to a more strict "two reliable sources" rule). I note that what Langus points out is also of relevance to establish the importance of such a section being present in the article.
@Irondome: the source you presented is a good one to give an idea of the weight assigned to the international position on the issue, it doesn't mean we should base the entire section on that article or only articles from that source (in fact: we should not, the more articles from reliable sources the better). The problem with attempting to come up with a summary like you propose is that it would be very difficult to come to an agreement. In this case, short mentions of individual (or grouped, as to avoid listing) positions could prove to be less prone to attract never-ending discussions.
This is what I suggest (all in one single section with separated paragraphs):

 International position (title)
 Opening paragraph about the disparity in pro-Argentinian and pro-British positions being voiced (properly sourced and neutral)
 UN resolutions (briefly, mainly: "The UN does not recognize the islanders as having a right to self-determination hence Britain continually rejects the repeated UN's calls for resolving the issue through dialog with Argentina")
 Mention of the US and EU neutral positions (the "Monroe Doctrine" should be moved to another part of the article as to not extend the US position too much) as major international players. The fact that the islands are listed as a British OCT by the EU and the Commonwealth should be mentioned too.
 Mention of the majority of Latin American states backing Argentina (as per article presented by Irondome, the article presented by me and this article accessible here, section 3.1). China should be added here too being another major international player.
 Finally Spain's position should be mentioned and also its similar issue with Gibraltar (although editors have previously opposed any mention of Gibraltar here, I still think it is worth of mentioning)

That's my proposal and I am of course open to hear comments and/or corrections. It should end up being quite shorter than the current section and easier to source too. Regards.