Rashford hit the post with his spot-kick in England's 3-2 shoot-out defeat to Italy – Gianluigi Donnarumma subsequently saving efforts from Jadon Sancho and Bukayo Saka.
United star Rashford had been introduced by Gareth Southgate in the dying embers of extra time at Wembley, seemingly specifically with penalties in mind.
After the game, the 23-year-old – whose fight for children to have access to free school meals has led to a shift in UK government policy – along with Saka, Sancho and Raheem Sterling, was targetted by hateful messages on social media.
Channel 4, quoting a report by data company Signify, claimed close to 2,000 racially abusive tweets had been sent to the four players.
Southgate, the Football Association (FA) and UK Prime Minister Boris Johnson all condemned the perpetrators, while England captain Harry Kane insisted anybody posting such messages was not wanted by the team as a supporter.
Meanwhile, a mural of Rashford in Withington, Manchester, was defaced overnight.
However, supporters of Rashford have since covered the graffiti with messages of support, while United's official Twitter account posted: "We're all behind you, @MarcusRashford.
"As a player. As a person. As an inspiration to our club and our supporters. As a representation of hope that there is plenty more good than bad in the world."
On Monday, Rashford posted an emotional message, firstly apologising for his missed penalty.
"I don't even know where to start and I don't even know how to put into words how I'm feeling at this exact time," Rashford wrote.
"I've had a difficult season, I think that's been clear for everyone to see and I probably went into that final with a lack of confidence. I've always backed myself for a penalty but something didn't feel quite right.
"During the long run up I was saving myself a bit of time and unfortunately the result was not what I wanted. I felt as though I had let my team-mates down. I felt as if I'd let everyone down.
"A penalty was all I'd been asked to contribute for the team. I can score penalties in my sleep so why not that one? It's been playing in my head over and over since I struck the ball and there's probably not a word to quite describe how it feels. Final. 55 years. One penalty. History.
"All I can say is sorry. I wish it had of [sic] gone differently. Whilst I continue to say sorry I want to shout out my team-mates. This summer has been one of the best camps I've experienced and you've all played a role in that. A brotherhood has been built that is unbreakable. Your success is my success. Your failures are mine."
He then went on to thank the well-wishers for their support, though reiterated he would never apologise for "who I am and where I came from."
“I've grown into a sport where I expect to read things written about myself. Whether it be the colour of my skin, where I grew up, or, most recently, how I decide to spend my time off the pitch," Rashford continued.
"I've felt no prouder moment than wearing those three lions on my chest and seeing my family cheer me on in a crowd of 10s of thousands. I dreamt of days like this. The messages I've received today have been positively overwhelming and seeing the response in Withington had me on the verge of tears.
"The communities that always wrapped their arms around me continue to hold me up. I'm Marcus Rashford, 23 year old, black man from Withington and Wythenshawe, South Manchester. If I have nothing else I have that."