The Manchester United forward was one of three players to fail to score in the shoot-out alongside Jadon Sancho and Bukayo Saka. All three have since suffered racist abuse on social media.
In a moving message on Twitter, Rashford said: “I don’t even know where to start and I don’t even know how to put into words how I am feeling at this exact time.
“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. 1 penalty. History. All I can say is sorry. I wish it had gone differently.”
Rashford praised the England squad for building an “unbreakable” bond during the tournament where the Three Lions made history by reaching the final of the European Championship for the first time.
He added: “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.”
The 23-year-old, whose mural in Withington was defaced after his penalty miss before being covered in supportive notes, promised he would bounce back from the disappointment.
“I’ve grown into a sport where I expected 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 said in reference to his campaign to help prevent child hunger in the UK.
“I can take critique of my performance all day long, my penalty was not good enough, it should have gone in but I will never apologise for who I am and where I came from. 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. For all the kind messages thank you. I’ll be back stronger. We’ll be back stronger.”
In response, the official England account responded to Rashford on Twitter with a message of support.
It read: “Keep your head up, Marcus. We know you’ll come back stronger from this. Continue inspiring. Continue making a difference. We’re extremely proud of you.”