In principal we were defending "democracy", but in reality we were just containing the spread of communism, which put our government in political bed with some pretty horrible leaders around the world. We were criticized in Europe and of course the Soviet Union for this as a hypocritical country that talked a good game when it came to democracy, but did not follow through in its foreign policy.
As for influencing Brown vs. Board of Education, there are some who believe the justices on the Supreme Court may have been influenced by the views of America around the world and the need to make a pro-civil rights statement, but the evidence on that score is pretty thin, and most of Africa was still a British or French colony in 1954. I'd say there's little connection between the two.
The US was the defender of democracy because we were trying to keep communism from spreading. We believed that we needed to keep the world safe for democracy by doing this.
One of the things we were trying to do was to keep communism from spreading into the Third World. Most people in the Third World are either black or brown. When the US had segregated schools, it made us look bad in the eyes of those non-white people.
So there was pressure on the Supreme Court to say segregation was illegal. That way, we would prove to the Third World that we were really committed to equality. This would make them more likely to come to our side in the Cold War.
The Cold War is the time period from 1945-1963. The United States was a key player in what was believed to be the prevention of the spread of Communism. It caused a series of events politically and socially that led to problems within the United Sates as well as world wide.
The Cold War began prior to the completion of World War II when armed forces worked together to over power Hitler's army and bring peace to Europe. Harry S. Truman and Joseph Stalin had opposing views.
Stalin wanted to create pro-Soviet governments through-out Europe. The United States industrialization and progress would help stabilize post war Germany and did not want Stalin's control through-out Europe. Stalin supported the development of the United Nations but was in controversy over all the other issues and ideas presented by Stalin. Stalin created the Eastern Block Wall to serve as a barrier between European commerce, The World Bank, and U.S. influence and the Western Block of Germany which was controlled by the communists.
Things served to get worse when the Soviet backed North Korea tried to over take South Korea. The United States felt that we needed to step in and prevent Communism from spreading through-out Asia. Military strength was increased.
Unfortunately the Red Scare became a common idea in America. Americans were afraid of the spread of Communism in our homeland. As a result people like McCarthy were able to persecute innocent Americans costing them jobs and creditability.
I have to allow my peers to decide about Brown Vs. Roe and the influence of the Cold War.