States tend to avoid war. Clausewitz once said that "war is a continuation of politics by other means", but the politics you must traverse before you can reach this point is wide, deep and extremely thorough. Both the UK and Argentina are members of the United Nations, and this is critical at this juncture. First, though not foremost, both are required to abide by Article 33 of the UN Charter, which reads:
1)The parties to any dispute, the continuance of which is likely to endanger the maintenance of international peace and security, shall, first of all, seek a solution by negotiation, enquiry, mediation, conciliation, arbitration, judicial settlement, resort to regional agencies or arrangements, or other peaceful means of their own choice.
2)The Security Council shall, when it deems necessary, call upon the parties to settle their dispute by such means.
The UK and Argentina have engaged in talks before to try and settle this matter, and I would not be surprised if they repeated this gesture in the near future to try and secure a peaceful outcome. They have both talked about referring the matter to the UN, though as the UK holds a veto on the Security Council, this may prove to be a politically difficult process to engage in. The UNSC can - and may well decide - to refer the matter to the ICJ, which both slows down the dispute and would allow for more neutral arbitration. I can't really see an outside power without enough of a vested interest to be acceptable to both sides. The UN system itself is replete with means to help solve this discussion. But it's important for other reasons.
The UN system represents an underlying set of values and beliefs in international society. Among them is the idea that states can no longer use force as a means to settle disputes. As outlined in Article 33 above, states have to exhaust all non-violent means to settle disputes. Even then, they can only use force if authorised by the UNSC (Article 42, unlikely owing to the UK veto) or through means of self-defence (Article 51). Though 51 has been read very broadly, in particular by the US over Iraq, where self-defence was associated with clairvoyance, it is extremely unlikely that Argentina - not a great power, let alone a super power - could get away with such a manipulation of the text.
Beyond this, states do not recognise the forceful occupation of the territory of another, it sets a very dangerous precedent for international relations. It is why Northern Cyprus is only recognised by Turkey (who occupied the territory) and is referred to as the 'Stimson Doctrine'. You cannot build an international society on principles around non-use of aggressive force and then recognise the forceful seizure of territory. Argentina is, again, not a great power and does not have either the means nor the norm innovative capacity to overturn this norm. Venezuela's backing is telling - they lay claim to a large portion of Guyana, and any change to this norm allow Chavez to try to seize this. However, it's very unlikely that a majority of states, certainly a majority in terms of power and norm considerations, would back an end to this new norm. Weaker states would object to the loss of a norm that protects them from attack, and stronger states would object - because they'd end up being dragged into the disputes of neighbours and allies.
War is expensive - coercion is seen as the least efficient means of enforcing new norms, or indeed achieving a result in foreign policy terms. Not only does the norm protect states, it also releases them from the costs of having to maintain a defence against all potential threats.
Argentina has nothing to gain from forcing its way into the islands, but the UK should not (and is not, I should stress) countenance an excessive response, as many of its less-able citizens have been urging. There is no need to make a great enemy out of Argentina, only to defend the islands against the unlikely event of an invasion and to reinforce the norm that you use enough force to defend yourself, but not so much that it becomes a matter beyond the purview of Article 51. The current normative structure suits both the UK and Argentina far more than an alternate system would, though some of Argentina's backers may be itching to get their hands on some of their neighbours through force. Nevertheless, the overall system is safe.
In summary, there will be no war. The UN system provides plenty of means to resolve the dispute peacefully and effectively, and the underlying norm structure suits both states enough to mean that they won't try to overthrow it. War is a very inefficient means of solving a dispute, even one involving lucrative resources, and would challenge deeply engrained norms in international society. Argentina has far more to gain from keeping the current norm structure than overthrowing it, and doesn't have the capacity to do so in any case.