Short answer: America HATED that Argentina was neutral for most of World War II.

Long answer: Argentina and America mutually hated and defied each other for basically the whole war and beyond, which, incidentally, is one of the things which led to Argentina's current (somewhat deserved, but not NEARLY as much as one would think) reputation for being "refuge of the Nazis." What exactly happened?

The US and Argentina already clashed in terms of their views on alliances and diplomacy. The US wanted a Monroe Doctrine-style, pan-American alliance- of which, naturally, it would be a leader. Argentina favored a more independent minded approach which emphasized its own stance as a trading partner with European countries and a world bread basket (as opposed to an ever more uneven trade imbalance with the US). Consequently, not wanting to be edged out of power by US hegemony over a potential hemispheric alliance, Argentina generally rebuffed US attempts to form an anti-Axis force as early as 1938. This isn't super surprising when one remembers the large Italian and German immigrant populations in Argentina, and Argentina certainly had some Germanophile leanings. But in fact Argentina's reasons for not joining had little to do with any such leanings, and more to do with not wanting to end up in a subservient stance to the US in some kind of hemispheric alliance. Argentina was actually working quite closely with Britain, and in fact some of Argentina's antipathy to the US had to do with that alignment.

Originally, from 1939-1941, the US and Argentina weren't quite so antipathetic- the president, Roberto Ortiz, showed pro-democracy leanings and a desire to work with the US somewhat. FDR's Good Neighbor policy, in theory, should have meant that American nonintervention in Latin American affairs would help maintain a good relationship while not being threatening to Argentina. But with an increase in nationalism in the early 1940s, accompanied by Ortiz ceding power due to illness and being replaced by Ramon Castillo, who was more antidemocratic and even less inclined to cooperate with the US, the US-Argentina relationship became more strained. In 1942, after the US had joined the war, they immediately sought to compel the Latin American countries to declare war as well, or at least to break off diplomatic relations with Axis countries- and Argentina refused to do so. The reasons for this were certainly linked to resentment of the US, particularly in that a) the US had leaked some initial considerations of Argentina in supporting the Allies and b) by the US refusing to support Argentina militarily in the way that it did the other Latin American countries. But it was also due to longstanding Argentinian policy regarding world war- Argentina had been neutral in WWI as well, and wanted to maintain its trade deals in Europe. In fact, Argentina was mostly helping the UK in the war effort more than anyone else.

Basically, this led to an onslaught of vitriol directed at Argentina by the US government (which also focused on supporting Brazil economically and militarily). Cordell Hull (ironically the creator of the Good Neighbor Policy") was one of the most notorious of these haters, and was described by Sumner Welles as having "an anti-Argentine bias that was almost psychopathic." He was part of a faction in US foreign policy which favored harshly punishing Argentina for not joining the war (and also for favoring British trade over American interests), and in fact the US treated Argentina terribly, far worse than any other neutral country including Francoist Spain, due to this antipathy which was propagated in the US. Argentina was seen as being the location of the future Fourth Reich, a Nazi stronghold, with a Fascist government which intended to invade Brazil, Bolivia and Uruguay to establish lebensraum. (While it is true that there were some people in the government with fascist leanings, and the Argentinian military had a legacy of pro-German and then pro-Nazi tendencies, obviously all of the above was far blown out of proportion.) Interestingly, the UK, Germany and Italy did not share this opinion of Argentina as being practically an outpost of Axis power, but that didn't stop the spread of this perception in the US. US public opinion of Argentina was in the toilet, and Cordell Hull was one of those about to flush it down.

This wasn't just internal policy or ideology- it led to very real ramifications for Argentina. Hull decided that Argentina should be pressured into joining the war against the Axis powers by whatever means necessary. This led to massive economic boycotts and sanctions against Argentina, which lasted until 1949, led to crippling effects on Argentinian trade in general with many other nations (including the UK), and can very much be considered part of the reason why the Argentinian economy degenerated in the mid 20th century from its relatively high point previously. Argentinian anger at this led to talk of the formation of an anti-US faction of Latin American countries, though this never really took off. Hull also disclosed Argentina's request for Lend-Lease aid, to Argentina's intense embarrassment, leading to the resignation of Argentina's foreign minister. Hull's immense pressure on the (antidemocratic) de facto president of Argentina, Pedro Pablo Ramirez, to join the war finally led to Ramirez deciding to break off diplomatic relations with the Axis powers. This made him vulnerable to the largely neutralist forces which surrounded him in Argentina, which led to his government being deposed in a coup d'etat in 1943, after which the presidency was taken over by Edelmiro Farrell and his right hand man Juan Peron (directly leading the way to Peron becoming the next president in 1946). While the US government originally counted this as a victory, they then refused to support Farrell when he didn't prove any more interested in joining the war than his predecessors had been. The US ambassador was withdrawn and the economic and diplomatic sanctions became even stronger.

It wasn't until March 1945 that Argentina finally declared war on the Axis powers, when, knowing that the war would be over soon, they signed the Act of Chapultepec, a treaty between the hemispheric nations (from the drafting of which Argentina had been deliberately excluded by the US) which declared that if one nation was attacked then all would rise to their defense. (This made Argentina officially the last country to join the war.) It also forced all nations to purge themselves of Nazi influences, which Argentina committed to doing as well. In return, the US formally recognized the Farrell-Peron government, though its antagonism didn't cease. Peron came to be seen as a symbol of Nazism, rather than the symbol of Argentinian nationalism and populism which he in fact was. (While he had some admiration for fascist leaders like Hitler and Mussolini, he was by no means a Nazi and in fact was one of the most Jewishly inclusive rulers of Argentina to that point, as I wrote in this comment.) This directly led to Argentina's reputation as a haven of Nazism, which, despite Argentina's acknowledged Germanophilia and the fact that it was in fact a main initial destination for many Germans and Austrians postwar, is an overblown assertion. For one thing, it ignores the fact that only a quarter of the 80,000 Germans and Austrians who emigrated to Argentina stayed; for another, it deflects from the fact that while Peron may have actively courted German scientists and industrialists to Argentina, other countries like the US, UK, USSR, etc did the same thing. While it's certainly a part of Argentina's very checkered past, US antagonism toward Argentina definitely made the Argentina-Nazi connection seem bigger and more eerie than it actually was.

Sources:

Escude, "US Political Destabilization and Economic Boycott of Argentina during the 1940s"

MacDonald, "The Politics of Intervention: The United States and Argentina, 1941-1946"

Trask, "Spruille Braden versus George Messersmith: World War II, the Cold War, and Argentine Policy, 1945-1947"

Greenberg, "From Confrontation to Alliance: Peronist Argentina's Diplomacy with the United States, 1945-1951"

Steinacher, Nazis on the Run: How Hitler's Henchmen Evaded Justice