This statement is indicative of the period in which the novel is set. While the Civil War in theory freed slaves and gave them American citizenship, they were considered second class citizens. They were denied the right to vote in most Southern states, and were subject to the infamous Jim Crow Laws. Immediately following the close of World War II, the African American population began a concentrated push for Civil Rights, which would ultimately give birth to the African American Civil Rights Movement that peaked in the 1960s.
In relation to Sophie's Choice, this is significant for a number of reasons. Nathan, a Jewish-American, will constantly come back to the theme of slavery in the American South. Nathan is driven by a need for justice at a time when many war criminals were going unpunished. And it is for this reason that he keeps coming back to this topic, and using it against the Southerner, Stingo.
Like many in American society during this time, Nathan felt that American civilization was one of the greatest examples of hypocrisy ever seen. While condemning the Nazis for their Jewish genocide, Americans continued to uphold racist ideals in their own country. Not only did they continue slavery after most other countries had abandoned the practice, but even following the Civil War they failed to correct the wrongs that had been perpetrated, permitting the continuance of brutal acts of violence, the enforcement of unjust and unequal laws, and the denial of the basic rights to the African-American population. It was these injustices that Nathan felt so strongly about, and it is for this reason that the issue of African-American slavery plays such an important role in Sophie's Choice.