The 23-year-old England international said he refused to take screenshots of the multiple messages he received as it would be irresponsible to do so.
"Humanity and social media at its worst. Yes I'm a Black man and I live every day proud that I am. No one, or no one comment, is going to make me feel any different," Rashford wrote on Twitter late on Saturday.
"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... 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."
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.— Marcus Rashford MBE (@MarcusRashford) January 30, 2021
The Football Association (FA) had earlier condemned the racial abuse of players after United duo Axel Tuanzebe and Anthony Martial, West Bromwich Albion's Romaine Sawyers and Chelsea's Reece James were targeted this week.
"We are united with all of football in our abhorrence of any racist abuse. This is not acceptable in any part of society," the FA said in a statement on Twitter on Saturday.
"We will continue to work with the rest of the game, the government and social media platforms to remove this - and all elements of - discrimination from our sport."
Police on Friday said they had arrested a 49-year-old man relating to the abuse of Sawyers.
Government ministers met top English players on Monday to discuss online abuse and discrimination as part of a series of "Future of Football" discussions.