Bob Jones/Bob Jones University is probably the most (in)famous defender of American racial segregation on the Christian side of things.
In 1960, Bob Jones Sr. issued a sermon entitled "Is Segregation Spiritual?" apparently issued in direct response to a call by Billy Graham for church leaders to get behind ending segregation. The sermon was then made into pamphlet form and published as a justification of the universities stance against allowing black students. Here is a scan of the pamphlet, which I will summarize below.
Jones' argument is based largely on a single Bible verse, Acts 17:26:
And he [God] made from one man every nation of mankind to live on all the face of the earth, having determined allotted periods and the boundaries of their dwelling place, (ESV)
To which he says
Now, what does that say? That says God Almighty fixed the bounds of their habitation... God Almighty did not make of the human race one race in the sense that He did not fix the bounds of their habitation. That is perfectly clear. It is no accident that most Chinese are in China.
He then suggests that racial conflict exists solely because man has disobeyed God's plan and moved out of their appointed territories. In fairness, he makes it quite clear that "superiority" of one race doesn't enter into the equation:
The Chinese people are wonderful people. The Japanese people are ingenious–they are wonderful people. The Koreans are wonderful people. The Africans are wonderful people. In many ways, there are no people in the world finer than the colored people who were brought over here in slavery in days gone by.
You talk about a superior race and an inferior race and all that kind of situation. Wait a minute. No race is inferior in the will of God. Get that clear. If a race is in the will of God, it is not inferior. It is a superior race. You cannot be superior to another race if your race is in the will of God and the other race is in the will of God. But the purposes of these races were established by Almighty God; and when man attempts to run contrary to the directive will of God for this world, there is always trouble. Now, that is the trouble.
Jones says white men broke God's will when they brought black slaves to America (duh), but God overthrew that evil plan and created a good one its place by ending slavery and making the former slaves into Christians, saying the black church is in many ways superior to the white church. He then claims the desire to integrate comes not from blacks, but from "Satanic agitation" using misguided white pity. And integrating will destroy not only the white church, but also the black church.
Jones then supplies some examples of "the races [mixing] up in large numbers" that caused problems. Again and again he says none of these things would have happened if people just stuck to their assigned habitation.
Then, Jones turns to his only other Biblical argument:
God Almighty had a purpose for the Jewish race; and for that purpose to be carried out, He had to separated them from among the nations of the earth. God chose Israel; and through the loins of Israel, He brought us the Messiah... [The Jews] have been scattered over the face of the earth, and they have kept their racial identity through the years.
The Jews are back in Palestine with a Government today. God scattered them, but He brought them back to their homeland...
Yes, God chose the Jews. If you are against segregation and against racial separation, then you are against God Almighty because He made racial separation in order to preserve the race through whom He could send the Messiah and through whom He could send the Bible. God is the author of segregation. God is the author of Jewish separation and Gentile separation and Japanese separation. God made of one blood all nations, but He also drew the boundary lines between races.
So, God created race, God authored the boundaries that separate the races, God segregated the Jews from all others and His salvation plan could not have occurred otherwise, and God wants the boundaries to remain (Acts passage). Therefore, segregation is God ordered, and integration is a "Satanic aggregation". But, all races are equal in God's eyes - they just have different traits and different purposes; they should remain separated so that each race can fulfill its intended purpose.
(It should go without saying, but I obviously do not endorse any of this myself.)