From the article David provided the link to

He feared that support for a European colonial power would undermine ties with Latin America and hamper Washington’s covert campaign against communism in the western hemisphere.
Our relationship with the UK and other colonial powers has a long history of being troublesome, and the deals we made to support France, UK, and the Dutch regain their colonies in Asia alone after WWII have resulted in tens of thousands being killed. The American idea of independence was compromised and we lost considerable influence that at the time played into the hands of the communists.

Fast forward to 1982 and Latin America where we were supporting a number of States battle communist insurgencies and had the Monroe Doctrine I can understand Haig's position. I definitely don't agree in putting an ally at risk by compromising their mission, but overt U.S. support couldn't be beneficial to our longer term interests in the region. Some other approach to mediate the conflict may have been possible, but I suspect like everything we do it was crisis action planning and they were looking for expedient means to minimize damage to our national interests.

We had a similiar spat where we actually acted (diplomatically and economically) during the Suez Crisis (1956/7) time frame against the UK and France. Nassar and the Egyptian people saw the UK as a greater threat to their national interests than the USSR, and the US needed Egypt in their anti-communist coalition. Nations act to pursue their interests.