Although many cultural values and norms changed during the Cold War period, not all of those changes were due to the Cold War itself, and social norms can rarely be considered settled, as they always change over time; there are few cultures which have the same social norms and values they did a few hundred, much less a few thousand years ago.
In World War II, soldiers from various different social and ethnic backgrounds fought together in the United States forces. Navajo code takers, black people, white people, and many other Americans were brothers and sisters in arms, something that broke down racial and ethnic barriers. Revulsion against the Holocaust made anti-Semitism increasingly unacceptable. These factors contributed to the civil rights movement. Having the USSR as a common enemy may also have made American society more cohesive.
The deeply unpopular Vietnam War was part of the Cold War and sparked many of the protest movements that defined the social unrest of the 1960s. The draft also led to protest movements, with students feeling that if they were liable to be drafted and sent to die, they should have more of a voice in elections and politics.
The Sputnik moment and returning veterans taking advantage of GI Bill educational assistance meant that many more people went to university. Well-educated people often have different social values concerning issues such as gender and race than people with less education.