Rashford, 23, became the latest United player to be targeted after Anthony Martial and Axel Tuanzebe were insulted earlier this week.
The England international described the abuse as society, and social media, at its worst.
"Humanity and social media at its worst. Yes I'm a black man and I live every day proud that I am," Rashford wrote on Twitter.
"No one, or no one comment, is going to make me feel any different. So sorry if you were looking for a strong reaction, you're just simply not going to get it here. I'm not sharing screenshots. It would be irresponsible to do so and as you can imagine there's nothing original in them. I have beautiful children of all colours following me and they don't need to read it. Beautiful colours that should only be celebrated."