"White moderate" isn't the name call. The implication in the quote is "MLK says this about you and he says you don't give a shit and you're possibly an enemy and possibly even racist yourself." On a liberal skewing message board, there's a loaded meaning behind the quote... and its not very hidden. This exact block quote pops up every single time there's any form of disagreement in any way with any aspect of a protest for the shaming reason you describe.
It's not even a means of saying someone view is wrong because of specific reasons, it's just invoking a venerated authority like MLK to put others - specifically whites - as enemies of that authority. In a discussion involving race, suggesting that a white person stands opposed to MLK, hell that MLK identified them as almost as bad as the KKK, is a "subtle" means of attacking their character.
Like this post below going for the "just asking questions" means of implying racism. At least here there's no subtlety.