The text of the Equal Protection Clause is:
"nor shall any State ... deny to any person within its jurisdiction the equal protection of the laws."
White American segregationists maintained that separate facilities (e.g. public schools, buses, etc) could be mandated for both blacks and whites, as long as the facilities were equal, and that the black facility is just as good as the white one.
Some integrationists arguments against this are: 1) There will be systematic bias preventing the black facility from ever being as good as the white one, 2) The motive of segregation is to keep whites away from blacks, not blacks away from whites, so the law favors whites, 3) The psychological impact of the law on blacks is worse than the psychological impact on whites.
For (1), no law is perfectly enforceable. And from what I read, they seemed to do an ok job in maintaining facilities for blacks equally as facilities for whites. So (1) seems not to have been a real issue except maybe in some exceptional cases.
For (2), motive does not matter, because just because whites and not blacks want a certain law does not mean that it deprives blacks of equal protection.
(3) is just a fact of life and it applies to all laws - any law will have a worse outcome for some people than others.
However, even though I think these arguments are not valid, I do still think "Separate but Equal" is a violation of the 14th Amendment. The reason is that protecting whites from blacks is not equal to protecting blacks from whites. Blacks are louder, more likely to commit crimes, are of a lower social class, so to protect whites from these people while not protecting blacks from people like this gives whites an unfair advantage.
I think if the integrationists had used this argument in Plessy vs Ferguson, they would have won and government-mandated segregation would have been struck down in 1896.