United Nations
The United Nations was formed out of the ashes of World War 2 as a means to prevent another world war. It replaced the League of Nations.
Speech Relevance
In 'A Time For Choosing' Reagan refers to an "International Organization" which must be the UN given the further reference to the "General Assembly":
I think we are for an international organization, where the nations of the world can seek peace. But I think we are against subordinating American interests to an organization that has become so structurally unsound that today you can muster a two-thirds vote on the floor of the General Assembly among the nations that represent less than 10 percent of the world's population.
His statement seems perfectly legitimate. The current Wikipedia article states that a 2/3 majority can be reached with 8% of the world's population.
Conceivably, the one state, one vote power structure could enable states comprising just eight percent of the world population to pass a resolution by a two-thirds vote (see List of countries by population). However, as no more than recommendations, it is difficult to imagine a situation in which a recommendation by member states constituting just eight percent of the world's population, would be adhered to by the remaining ninety-two percent of the population, should they object.
At the time of the speech, the United Nations had 110 members (Kuwait was added May 14, 1964), so 74 countries would have been required for a majority. The approximate population in 1964 was 3.276 billion. 10% of that population is 327.6 million, less than the population of China or India at the time.
