What are the similarities between the U.S. at the time of Lincoln's 2nd Inauguration and South Africa at the time of the Apartheid laws?

I think that you are probably examining how Jim Crow
legislation in the South is similar to apartheid legislation in South Africa.  There is
much in way of similarity.  White Southerners did not want to give rights to newly freed
slaves in much the same way that Afrikaners did not want to give rights to Black South
Africans.  Both used the law as a way to maintain control over their people and made
sure that there were strict penalties for those who violated the stratification present
in the legal code.  Both apartheid and Jim Crow laws were examples of substantive due
process violations, where the law itself was wrong and violated individual rights.  In
the end, both segregation and apartheid only met their end when intense social pressure
came to a boil and forcibly dislodged them out of practice.