Having come through a group containing Belgium, Italy and Sweden, Ireland were dumped out of the tournament at the last-16 stage by hosts France after losing 2-1 at Parc Olympique Lyonnais on Sunday.
After the defeat, the Irish players and staff remained on the pitch to applaud the fans who made their way to the game and the Everton defender stressed the importance of that "special" relationship.
"Our fans are immense and we're so proud of them, as I'm sure they are of us. It means a lot to us - not just the fans who were there today, but the fans who were at the Sweden game, the Belgium game and the Italy game.
"They all played a big part. And all the fans back home watching, there is that bond. It would have been nice if we could have gone one further."
Ireland drew their opening game against Sweden before losing to Belgium, but a victory against Italy in their last group game ensured progression to the latter stage of the competition.
Despite their tournament coming to an end against the hosts, Coleman believes that the performances of the younger players in the team indicates that "the future is bright" for Irish football.
"We were probably written off a bit after the Belgium game [3-0 defeat], but I'm sure you've seen a few good young lads come through," the former Sligo Rovers defender said.
"The likes of Jeff [Hendrick], Robbie [Brady], James [McCarthy], big [Shane] Duffy as well. [They] played a big part in this tournament.
"I suppose the future is bright in terms of those young lads coming through."
Ireland now turn their attentions to qualifying for the 2018 World Cup in Russia and their campaign gets started on September 5 away to Serbia.