He was went off early in City's Champions League semi-final second-leg defeat by Real Madrid on Wednesday.
City boss Manuel Pellegrini said the defender would undergo surgery and may be out "for a long time".
Kompany, who has had five injuries this season, playing just 22 matches out of a possible 58, said it was "very sad news" but he would not "give up".
His statement added: "I trust my friends and team-mates to do well and do us proud at the next European Championships in France.
"I am now officially their most fervent supporter and I will be there to support them."
Belgium are grouped with Italy, Sweden and the Republic of Ireland at this summer's tournament.
Chelsea's Eden Hazard is Belgium's vice-captain and has led the side in previous games when Kompany has been unavailable.
Kompany also wrote he would "remain calm and positive" rather than "feeling sorry for myself".
He added: "I could fear for my career, as others may do. I could give up. That is not me. I decide on my future."
Belgium and Manchester City captain Vincent Kompany will miss Euro 2016 with a groin injury.