Steve Clarke’s side fell to a 3-1 defeat to Croatia at Hampden Park which left the team bottom of Group D and out of the competition.
It brought an end to Scotland’s first tournament in 23 years.
He said: "Where to even start?! I suppose the first thing to do is to say that we are sorry for falling short in the end. We desperately wanted to make even more history but we didn’t quite get there and it hurts.
“Seeing the entire country not only following what we’ve done and supporting us but lifting us every step of the way made a big difference.
“From the incredible support of the Tartan Army to the kids in the street wearing Scotland shirts,we’ve felt your love and been inspired by it
“To captain this country and these boys means the world to me and the feeling I’ve had leading them out in a tournament is impossible to describe!
“What matters now is that we do all that we can to ensure that this is the start of a wonderful journey rather than the end.
“We want more. We want to qualify for more tournaments and we want to make the people of this magnificent country even prouder!
“We will give everything we have to make this a reality.
“Scotland it’s been a pleasure.”