England international Rashford tweeted after the game that he had been the focus of racial abuse on his social media accounts.
“He teaches children!! And knows that he can freely racially abuse without consequence…”
Manchester United tweeted that it was aware that some of its players had been the subject of racist abuse.
A number of players in English football, including several of Rashford’s teammates, have also been targeted by online discrimination in recent months.
In February, racist symbols were left under an Instagram post by Anthony Martial following United’s 1-1 draw with West Bromwich Albion.
The Premier League, in unison with the rest of English football, held a social media boycott from April 30 to May 3 to bring awareness to the racial abuse directed at players.
In Wednesday’s final, it was Villarreal who took the lead just inside the half-hour mark when Gerard Moreno connected with Dani Parejo’s free-kick — the first goal scored between these two sides after their previous encounters produced four goalless draws.
That was Moreno’s 30th goal of the season and saw Villarreal lead at halftime. United did get an equalizer in the second half when Edinson Cavani reacted quickly following Paul Pogba’s blocked shot from a corner to poke the ball past Gerónimo Rulli.
Despite enjoying more chances throughout the second half, United was unable to take the lead, and the game went to penalties after an uneventful extra-time.
The penalty shootout, however, did provide drama.
The teams went toe-to-toe throughout before De Gea’s penalty was saved by Rulli, handing Villarreal a 1-1 (11-10) victory.
It was the sixth time in the past eight competitions that a Spanish side has won the Europa League, European football’s second-tier competition.