United was beaten 1-0 in the second leg of their round-of-16 tie against the Spanish side on Wednesday (AEDT), losing 2-1 on aggregate.
The video posted after the game appeared to show Rashford confronting a supporter who had heckled him outside the stadium, and the England international has denied that he raised his middle finger to fans.
Rashford posted a message on Twitter on Wednesday along with the words "There are 2 sides to every story."
There are 2 sides to every story. pic.twitter.com/Xl2PRyaX2c— Marcus Rashford MBE (@MarcusRashford) March 16, 2022
He wrote: "A video can paint a thousand words and in this case lead to inaccurate info being shared on social media.
"Guys, for weeks I've been heckled, threatened, questioned and last night my emotion got the better of me. I'm a human being.
"Reading and hearing that stuff about yourself every day, it wears you down. No one is more critical of my performance than me. But what you see in this video lacks context. I had been heckled from the minute I stepped foot outside the ground, abuse not just aimed at my football.
"People were looking for a reaction from me. Phones were at the ready. Of course, I should have walked straight past and ignored it, that's what we're supposed to do right?
"I want to clarify two things. The first being what I actually said to the man throwing abuse at me which [was] 'come over here and say it to my face' (a fact security can back up) and secondly, the fact I used my forefinger to direct the fan to 'come over and say it to my face'. I did not gesture with my middle finger.
"I'm not entitled. This isn't ego. I'm upset. I'm disappointed. And in that moment it was silly but I was being human."
Rashford has struggled for form this season, scoring five goals in 26 appearances in all competitions (15 starts), and came on as a 67th-minute substitute in the Atletico defeat.