How did Americans respond to the challenges to the social order presented by their doctrine of equality?

This is a broad question, but over the course of their history, Americans with power (i.e., whites) have responded to the conflict between the equality promised in foundational documents of the United States, such as the Declaration of Independence and the Constitution, and the existence of an unequal social order, especially in the South, by reaffirming pre-existing racial and gender categories. In other words, instead of promoting widespread equality across the board at the beginning of the republic, elites made sure "equality" became the exclusive domain of white males. Groups such as black slaves, Native Americans, and women were excluded from notions of equality.
Native Americans, for example, were to be allowed sovereignty over their reservations only after they had given up their "primitive" ways and adopted white cultural norms. Until then, they were under rigid supervision by white government superintendents.
Black slavery was first upheld. Then, even after emancipation, segregation, which denied blacks equal access and equal opportunity, was normalized for a century. United States history has been characterized by a long struggle by people considered non-white or non-Nordic, such as Eastern or Southern European immigrants, blacks, Asians, and Native Americans, for legal equality. Women, too, have had to struggle for full legal personhood and the franchise.