Hazard, who spent seven successful seasons at Chelsea before joining Real Madrid as a record signing in the summer of 2019, issued an apology on Instagram on Thursday.
He wrote in both English and Spanish: "I am sorry. I have read lots of opinions about me today and it was not my intention to offend the Real Madrid fans. It has always been my dream to play for Real Madrid and I came here to win. The season is not over and together we must now battle for La Liga! Hala Madrid!"
Hazard, 30, started in the semifinal return leg to lead Madrid's attack alongside Karim Benzema but was anonymous throughout the game and was replaced by Mariano in the 88th minute with his team losing 2-0. Madrid lost the tie 3-1 on aggregate as Thomas Tuchel's team moved on to the final.
Madrid fans took to social media to criticise Hazard's reaction, and ESPN sources said that the "monumental anger" has extended to several club members and even the changing room.
Hazard has played only 37 games in two seasons at Madrid and has suffered 11 injuries since his arrival.
Certain Spanish media suggested that his laughs after the team's elimination by Chelsea should bring about the end of his career at Madrid.
Real Madrid captain Sergio Ramos, meanwhile, also spoke about his team's need to bounce back after seeing the club's dream of winning the European competition for the 14th time dashed.
The veteran defender wrote: "We're disappointed, yes. But we're not finished.
"A history built on victories but #realmadrid has always risen from defeats. The league title is still there for us and we're up for it."