Acts 17:26 can be, but is not necessarily, used as a justification for racial segregation. Since most people think that segregation is a result of and a form of racism, it is possible to argue that this verse can be used to say that racism is sanctioned in the New Testament.
The text of Acts 17:26 is as follows:
And he 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
We can argue that this verse could actually be used to argue against racism. It says that God made “every nation of mankind” out of “one man.” This would indicate that were are all brothers and sisters (or at least related) because we all came from the same ancestor. If we all came from the same ancestor, we should have no reason to be prejudiced against one another.
However, this verse can also be used to support and defend segregation. It says that God set out different places for each race to live. This implies that God did not want the races to be together in the same place. If God does not want the races to be together, then it would seem to be right for Christians to support segregation. If segregation is racist, then (by this understanding of the verse), Acts 17:26 supports racism.
We can see in the link below that at least some people who support the separation of the races use this verse to justify their beliefs, though they would claim that they are not actually racist for espousing these segregationist beliefs.
It could be used by saying that everyone started out from the same bloodline and we as human are no different now. No race is better than the other, they just have more distinct features. Color doesn't make you who you are, it is your character. Many people look on the outside, but God looks at the heart.