A spectator in the Chelsea supporters' end was seen making a monkey gesture during the home side's 5-4 extra-time victory over United last month.
But after reviewing video footage and taking witness statements, police and prosecutors decided to drop the case because of "insufficient evidence" required for a prosecution.
The Crown Prosecution Service said in a statement that "to bring a charge for a racially aggravated public order offence we need to be able to prove in a court either that an individual's gestures demonstrate hostility towards the victim based on the victim's (race)."
The 28-year-old man had been banned from Stamford Bridge pending the outcome of the investigation.
"We're currently reviewing the situation," Chelsea said.