A video was posted by Icelandic media with the pair hanging out with girls in their hotel room on Sunday.
In a statement on social media, Foden said: "Following the story which has emerged today regarding my actions in Iceland, I want to make a full apology.
"I apologise to Gareth Southgate, to my England team mates, to the staff, supporters, and also my Club and my family.
"When I was called up by Gareth for these games, my first reaction was that of immense pride. To pull on that short for the senior team in my England debut was an incredible privilege.
"I am a young player with a lot to learn, but I am aware of the huge responsibility I have in representing Manchester City and England at this level.
"On this occasion I made a poor decision and my behaviour didn't meet the standards expected of me.
"I breached COVID-19 protocols put in place to protect myself and my England colleagues. As a consequence I will now miss the opportunity to travel to Denmark with the squad, and that hurts.
"I will learn a valuable lesson from this error in judgment and I wish Gareth and the team good luck this week."