The 18-year-old said he had learned a lesson from the incident, which resulted in his being dropped from the squad for Tuesday's Nations League match in Denmark, along with Manchester City midfielder Phil Foden.
England manager Gareth Southgate refused to comment on the nature of the breach but media reports said the pair left the bio-secure bubble to meet two local girls.
Both players have been fined 250,000 Icelandic kronor ($1,800, £1,350) by Reykjavik police and the Football Association said it would be launching a full investigation.
"Having now had the chance to reflect on what's happened, I can only apologise to everyone for the embarrassment I have caused," Greenwood said in a statement.
"It was irresponsible of me to break the Covid-19 protocols which are in place to protect players, staff and the public.
"In particular, I want to apologise to Gareth Southgate, for letting him down when he had shown great trust in me."
Greenwood added: "Playing for England was one of the proudest moments in my life and I only have myself to blame for this huge mistake.
"I promise my family, the fans, Manchester United and England that this is a lesson I will learn from."
Foden, 20, issued an apology on social media on Monday, saying he would learn a "valuable lesson".