I think you have to be careful with the vocabulary here. If a white man is being abused online, and he complains about it, his complaint may be rendered illegitimate on the basis of being "tone policing". He may be told that "freedom of speech does not mean freedom from its consequences". He may be told that the people abusing him are under no obligation to educate him and are doing "unpaid emotional labor".
"I call out bad behavior, they are trolling, you engage in targeted online harassment."
If anything, I've noticed that women and minorities seem to have more of a blank check to get really crass and say things like "go fuck yourself", etc. I see people being mocked on the basis of being "mediocre white men" far more often on Twitter than I see them mocked on the basis of being "mediocre women of color". Gendered/racialized equivalents of "toxic masculinity" and "whiteness" aren't as widely employed.
(Agreed that social media can be terrible; I hope this initiative makes it better.)