As I have little else to do today, I'll take these arguments on.
The Malvinas are within "walking distance" of Argentina (480km), sitting on top of the Argentinian continental shelf.
1. International law regards various offshore distances as being of significance to territorial disputes: three miles, seven miles and twenty-one miles, with two hundred miles as an absolute limit. Territorial contiguity (via an undersea continental shelf) does not seem to have much force in international law; otherwise presumably the Canaries would belong to Morocco.
The islands were discovered by Spain before the alleged discovery by John Davis in 1592, during a time when mere discovery granted rights of domain.
2. See the next point.
Upon independence Argentina inherited the right of the islands from Spain according to the doctrine of "uti possidetis iuris" and the succession of states, thus Argentina exercised "eminent domain" as early as 1810.
3. In March 1848, the delegates at the Lima Congress in Peru signed a number of agreements. Article 7 stated " The confederated Republics declare that they have a perfect right to the conservation of their territories as they existed at the time of independence from Spain, those of the respective Viceroyalties, captaincies-general or presidencies into which Spanish America was divided."
Article 7 was signed by Ministers from Colombia, Chile, Bolivia, Ecuador, and Peru. Argentina did not sign. Britain was already in control of the islands. Uti Possidetis Juris does not apply therefore to the Falkland Islands.
The UK occupied the islands by force in 1833, violating the territorial sovereignty of Argentina and evacuating it's inhabitants and never allowed them to return. This violates the UN resolution 1514 regarding the granting of independence to colonial states and people, which states in the 6th paragraph that any attempt at destroying the national unity and territorial integrity of any country is incompatible with the purposes and principles of the United Nations Charter.
In 1833 two naval vessels were sent by the United Kingdom to re-assert British sovereignty (which had existed since 1765) over the Falkland Islands, after the United Provinces of South America ignored British diplomatic protests over the appointment of Luis Vernet as governor of the Falkland Islands and a dispute over fishing rights.
As for 1514, the principle of de-colonisation can't really apply to the Falkland Islands, because the Falklanders do have the right to self-determination and indeed, subsequent resolution 2065 noted "the existence of a dispute between the Governments of Argentina and the United Kingdom concerning sovereignty over the Islands", and invited those governments "to proceed without delay with the negotiations... with a view to finding a peaceful solution to the problem, bearing in mind the provisions and objectives of the Charter of the United Nations and of General Assembly UN Resolution 1514 (XV) and
the interests of the population of the Falkland Islands."
In 1774, the UK abandoned the islands and signed a treaty that renounced all of it's claims. While Argentina has never faltered on it's claim of sovereignty over the islands (much less ceded the islands in a treaty).
The Spanish expelled the British colony in 1770, but it was restored in 1771 following British threats of war over the islands. However, in 1774, economic pressures leading up to the American Revolutionary War forced Great Britain to withdraw from many overseas settlements. By 1776, the British had left their settlement, leaving behind a plaque asserting British sovereignty over the islands. Britain did not abandon the claim and I think the treaty being referred to is the Nootka Sound Convention which cannot apply to the Falkland Islands. The British have never signed a treaty giving away the islands.