Conservatism is about upholding cultural tradition and being cautious about change. Civil rights is the opposite of this idea.
I addressed these semantics recently, but I'll do it again. Don't just use conservative and right wing synonymously. Just because we are in the US with a mostly capitalist tradition, the words get intertwined. There are radical right wingers even in right-wing countries. The situation only gets worse (more confusing) due to the 2-party system and current coalition consisting of free marketers (or sometimes more appropriately corporatists - take your pick) and white Southerners (some Christians, some not) who couldn't care less about economic policy so they vote by Party. They only care about upholding traditional values as they always have. Despite having nothing in common, these two groups are called right wing because they vote for the more capitalistic economic policy no matter what their real views are.
If the case had been a traditionally left-wing nation that otherwise contained some sort of a racial hierarchy for most of its history, would conservative refer to both of these things? Yes because it would be upholding the socialist tradition and the racial hierarchy. There could be radicals on both the left and right but conservatives likely only on the left. Radicals on the right who opposed civil rights would be the example of someone who is conservative on just an issue. Conservatives on the left who want civil rights would be liberal on a single issue.
I know that still makes conservative the term for racist, but I'd rather not even ponder a revolutionary who destroys a society with a great tradition of equality. I suppose that's possible as well.
The meanings of these terms are not consistent across cultures and hardly even across time periods within a country which is why I find right-wing and left-wing so important. Can you argue for the inclusion of race on the spectrum within those constraints? Possibly, but I still think Rockefeller has the strongest argument thus far. It's a conservative tradition that we are debating in a right-wing nation, but there were activist advocates against it from both parties. Does being radical on one issue in a RW country make you LW on that particular issue if you are agreeing with the LW radicals as King wants to portray (with terminology being my own)? Frankly, I'm not 100% sure - I tend to say no but its possible. Not every issue can really line up neatly into RW/LW lines. This is especially true when there is an underlying economic argument from your particular wing's ideology that explains why this cultural change would be best. That's not to say "Because it is would help the economy under our wing's conditions" but to say "Because the economically fair thing would be for the government to provide for this race as well" or "Because the economically fair thing would be to afford this race the same opportunity to produce and sell goods" or something more or less extreme from each side.
I know there's no definitive answer in there so this is a lot of rambling about word definitions, but I think Rock's more accurate than he has been given credit for.
And don't think I am just using conservative/radical to refer to cultural issues because there could easily be a right wing radical on an economic issue in a RW country.