While the "why" is more difficult to objectively answer, since it requires understanding the motivation of those who likely don't want their true motivations known, it is unfortunately extremely easy to answer the first question.

There was absolutely not a moral awakening in the South following WW2, and in fact, violence against blacks was specifically and especially targeted against service members. The Equal Justice Initiative compile what is probably the most thorough accounting of racial violence in America, counting nearly 5000 lynchings between 1877 and 1950 in their report "Lynching in America. They released an addendum a few years ago examining violence targeted at black veterans, which claimed "no one was more at risk of experiencing violence and targeted racial terror than black veterans".

This is probably best exemplified by the case of Isaac Woodward, who was beaten until blind while still in his service uniform for the temerity to ask a white bus driver if he could use the restroom at a stop. Stories like this were common, for instance black veteran J.C. Farmer was murdered for laughing at a bus stop while in uniform.

There was an idea in the South that black veterans were the most dangerous subgroup within the black population. They were afraid the respect and measure of equality military service had given them would make them more likely to seek equality in society. While he was speaking about WWI, Southern opinion of black servicemen still generally followed the thinking of Senator James K Vardaman:

It is a lamentable fact, and one we should be prepared to meet, that one of the horrible problems which will grow out of this unfortunate war, which the southern white people particularly must meet and overcome, is the training as a solider which the negro will receive. Impress the negro with the fact that he is defending the flag, inflate his untutored soul with military airs, teach him that it is his duty to keep the emblem of the Nation flying triumphantly in the air-it is but a short step tot he conclusion that his political rights must be respected, even though it is necessary for him to give his life in defense of those rights, and you at once create a problem far-reaching and momentous in its character.

This attitude was on full display following WW2. Many of the most prominent opponents of the civil rights movement were WW2 veterans. For example, Strom Thurmond fought with the 82nd at Normandy, and participated in the invasion of Germany, so likely almost certainly had direct exposure to at least some aspects of the holocaust.

For the rest of the country, however, the beating of Isaac Woodward was shocking. He was taken on a speaking tour of the US by civil rights activists, and was a powerful force in convincing Truman to desegregate the military (though he was also partially motivated by embarrassing propaganda from East Germany and the USSR, which used the segregated military units in West Germany to demonstrate the hypocrisy and inequality so fundamental to American society at the time. Desegregating the military also didn't stop the Southern lawmakers from systematically defunding organizations set up to ensure military equality, and denying GI Bill benefits to hundreds of thousands of black veterans.

I will refrain from speculating as to why the holocaust didn't encourage more Southern white veterans to push for racial equality.

Sources:

A Breath of Freedom: The Civil Rights Struggle, African American GIs, and Germany. Journal of Social History, Volume 46, Issue 1, Fall 2012, Pages 249–251, https://doi.org/10.1093/jsh/shs011

Williams, Chad L. Torchbearers of Democracy: African American Soldiers in the World War I Era. University of North Carolina Press, 2010. JSTOR, www.jstor.org/stable/10.5149/9780807899359_williams.

Lynching in America: Targeting Black Veterans. Equal Justice Institute, eji.org/wp-content/uploads/2019/10/lynching-in-america-targeting-black-veterans-web.pdf.

GERGEL, RICHARD. UNEXAMPLED COURAGE: the Blinding of Sgt. Isaac Woodard and the Awakening of President Harry s.... Truman and Judge j. Waties Waring. PICADOR, 2020.

u/limukala has covered this very nicely, but I would add that the question has an embedded premise that's somewhat anachronistic: "American soldiers who saw the racist atrocities of the Holocaust" -- not many soldiers saw the camps, and while some of the soldiers who _did_ see the camps viewed them as "racist atrocities" -- how far did that understanding permeate culture? Only a bit.

That they were fighting the Germans and that the Germans were killing people-- the soldiers already knew that. "German war crimes" was a well known and understood idea, for a generation since WW I, when German atrocities in Belgium and the sinking of civilian vessels were notorious. The racist nature of the Nazi genocidal project became clearer at Nuremberg and other trials, but if if you look at American popular opinion at the time, the Japanese were far more likely to be associated in the public mind with "atrocities" than the Nazis.

Of the two fronts, the Pacific War was the one understood by both soldiers and civilians as the more brutal. The "camps" that got much of the attention from the public at the time were the Japanese POW camps; that would have been on soldiers' and the public's mind. German behavior to Allied POW’s had been mostly correct - the Japanese had not. The soldiers who saw the German camps did come away shocked by the extent of the brutality, but this was seen as a Nazi atrocity first and foremost, and the racial aspects of it were much less often mentioned. Far from convincing Americans that we were wrong in our behavior at home-- the most notable moral result was to convince Americans that the war had been morally right; Eisenhower calls his memoir "Crusade in Europe" and that's a choice of words that made sense to his readers.

So very generally, it's hard to find a racist American soldier who had a "moral awakening" to segregation from exposure to the camps. Dwight Eisenhower famously inspected Ohrdruf Concentration Camp -- did it change his views on segregation? He was genuinely shocked and it did affect him as a moral human being, but I don't see him making the connection you imply. If you look at his letter to Marshall at the time, Eisenhower is appalled, but doesn't mention racial issues. The same impression can be seen in Patton's diary entry.

I'd say that the most consistent impact that military service had on American racial opinions was on African American servicemen-- Medgar Evers served in France, Charlie Evers in the Philippines for example; both brothers are said to have had relationships with women who'd they'd have been forbidden to marry in the US. So the war gave them and other African American servicemen the personal experience "things can be different"; that was much likely more influential in ideas about segregation than the camps. Similarly, African-Americans were enraged when they were held as inferior to German POW's-- prisoners could eat in "whites only" cafeterias, for example.

Even during the War, there was pressure on the Roosevelt Administration from African Americans that they were owed something better for their service-- the so-called "double V" campaign, that we're fighting against fascism abroad and racism at home. In 1948, President Truman signed Executive Order 9981 banning racial or religious discrimination in the US Armed Forces, although it took some years to fully come into effect, this effectively ended segregation in the military.

There are some good answers above already, but I want to delve into some of the underlying attitudes and ideals upon which soldiers and other Americans built their understanding of the camps. Then, we can look at how those perspectives were shaped postwar.

Part 1:

Great Question! It gets at some very interesting aspects of history and allows us to get at how people of the past viewed events through which they lived so differently than we do today. Namely, the American soldiers who liberated or viewed the camps didn’t see them as evidence ONLY of racial hatred. Certainly, they saw racism as an important part of them, but not the whole. The camps, in their minds, much just as or even more likely made them think of “divide and conquer” tactics, Nazi repression of Germans, anti-Christianity, and anti-trade unionism. This is because our contemporary view of the Nazis has been shaped by changing understandings of the Holocaust while theirs had not. The ubiquitous American soldier of 1945 had likely formed their understanding of Nazism in the years prior to the war.

What does all that mean? I am going to look at American educators to give an idea of the perspectives of Americans at the time.

“Divide and Conquer” & Subversion

Between 1933 and 1941, American educators perceived of Nazism (and Communism) as a threat to the Christian, democratic ideals which Americans held dear. In the midst of the Great Depression, they expressed real anxiety over the possibility that democracy might collapse from its internal problems. Further, as the ‘30s wore on, they perceived of Nazi subversion as a real peril that might undermine America and make it a fascist bastion. Though few at the time suggested an invasion by Germany, they did believe that Germany might attempt to weaken American democracy in order to topple it.

Take, for example, numerous teachers wrote of fear regarding the apparent ease at which other educators accepted totalitarian methods and ideology. In 1939, Saul Israel and Julia Speigelman wrote of “a sense of the losing battle which democracy seems to be waging throughout the world against the forces of totalitarian dictatorship.” In the same year, one of their fellow New York City teachers stated that “the frothy, sweeping, and seeming successes of Fascist countries and their ideologies, have so alarmed many of our friends of Democracy, that it has caused them to lose all faith in the power of their own ideology…these gentlemen and ladies have become so panicky that, in sheer desperation, they are ready to grasp at a straw.” These educators believed that Americans, in the midst of economic crisis at home and viewing the geopolitical one abroad, might turn to totalitarian systems in order to meet the threat.

The most common ideology to which educators sometimes turned, especially before the 1939 revelation of the Molotov-Ribbentrop Pact, was communism. Many educators saw its promise of social and economic stability as solving the problems in America. Marian Church condemned such vacillation on ideals. He noted that “a great body of our people began to doubt the sufficiency of our form of government to meet the situation. Some suggested that we needed a dictator like Mussolini to lead us…” while others favored “Communism in order to decentralize wealth in our country.” He, thus, recognized that “democracy was being tested. The supreme trial was at hand to determine the capacity of our people to participate in the government of their fathers.” Church argued for remaining dedicated to democracy.

Nevertheless, as the rumbling of war and, ultimately, war itself began in Europe, American educators felt more than that some Americans moved too far toward totalitarian ideologies. Many began to see signs of attempts at subversion. Though not unaware of communist subversion, the aggression of Germany made more look to Nazi attempts to undermine the United States.