If by support for the military you mean support for the United States to maintain a large military and use it to promote American power abroad, then the nadir would have to be the period between the end of the First World War and the start of the Second World War. After the First World War, the Republican-led Congress refused to join the Wilson-inspired League of Nations, but this was just the start of American isolationism and anti-alignment sentiment in the United States. By the 1930s, many began to speculate (perhaps “accuse” is a better word) that the arms industry had tricked the American government into joining the war in order to make a profit. The Senate even investigated the role of the arms industry in the American entry into the war via the so-called Nye Committee in 1936, which employed the then-popular term “Merchants of Death” during the investigation. The Committee did not find evidence of this conspiracy, but this did not end American distaste for war at the time. Around the same time, Congress also passed a series of Neutrality Acts (between 1935 and 1939) aimed at preventing the United States from once again becoming tied to European powers in the case of war (during the First World War, the United States had sold quite a bit of armaments to France and Great Britain on credit, which provided a strong incentive for the United States to enter that war. And remember, in the 1930s, the consensus was that the arms industry played a crucial role in the American entry into the war, and so these acts had wide support on these grounds). Once France fell to Germany and Britain stood as the last bastion of democracy in Europe, however, American sentiment had shifted enough that FDR was able to pass through Congress the Lend Lease Act of 1941, which provided the United States a loophole around the Neutrality Acts, and the US once again provided Britain with armaments to fight Germany. Needless to say, after the attack on Pearl Harbor, American sentiment radically shifted towards support for a strong American military, though this is not to say that anti-war sentiment disappeared from the country entirely (because it did not).

Someone might come in here and say that the anti-war movement of the 1960s-1970s represented the true nadir in American support for the military. But this is certainly not the case. For while many Americans vocally advocated for an end to both the Vietnam War and the costly arms race with the Soviet Union, by then the necessity of a strong American military had become a fact in the American mind. With the rise of the Soviet Union and its arsenal of weapons that could be used to conquer Europe or to attack the United States itself, Americans recognized that they no longer had the luxury of isolation that the Atlantic and Pacific Oceans had provided them during the interwar years. The sentiment was no longer that the United States could best serve its interests by maintaining a small military, and indeed the United States maintained a peacetime force for the remainder of the twentieth century far greater in cost and size than anything it had prior to the Second World War.