The Scots failed to qualify from Group D after a 0-0 draw with England and defeats by Czech Republic (2-0) and Croatia (3-1) resulted in Clarke’s men being the only team in the pool not to progress to the knockout stage.
Scotland who were the fourth seeds in the pool, were just one win away from progressing into the last 16 for the first time in its history and the disappointment of not doing so was clearly portrayed by Clarke in the aftermath of the loss to the Croatians at Hampden.
Carver, who has been a good friend and colleague of Clarke for many years, says the manager should be proud of the job he has done with the national team but accepted that his friend will need time to get over what happened at the tournament.
“Listen, when you're the main guy you take all the pressure on your shoulders,” Carver said of Clarke.
“It's easy for myself, Steven [Reid, coach] and Woodsy [Steve Woods, goalkeeping coach] to not feel it as much. We all feel pressure because we're part of something big.
“But it's also our role to be there for the manager. I wouldn't say it as a shoulder to cry on but certainly someone for him to lean on. Because he has to try and get over the disappointment. When you're the head guy, you feel it more than anybody.
“You take responsibility for it. He's got to be proud of what he's done. Because we almost did it but were just short in the end.
“Steve knows we're getting better and stronger. And we'll be better for the experience.
“I said I'd give him a few days more before we have a catch-up conversation.
“The manager keeps everything close to his chest. But I know him and he's hurting, of course he is. The day after the Croatia game, I could sense it in him. But it's my job to pick him up because we have a friendship as well.
“Steven goes back to Nottingham Forest, Woodsy goes back to Celtic – so I'll be there for him. We've been friends since 1999 so we'll have a friendship chat as well. He was definitely hurting the next day and he had to deal with that disappointment.”
Carver says the Scotland coaching team’s job is now to prepare for September’s World Cup qualifiers against Denmark, Moldova and Austria.
“I'm sore as well because I wanted to do well for Steve and the country,” continued Carver. “But we have to deal with it in a positive way and get ourselves ready for September.
“We'll have two or three weeks of down time and then get our heads on the World Cup qualifiers.
“I genuinely think the team has momentum. The nation has momentum.
“Although it was disappointing that we didn't qualify, the response from the fans after Croatia was fantastic.
“Remember, I was watching from afar the reaction when Scotland were losing to the likes of Belgium and Russia at the start.
“But there's been a bit of pride brought back to Scottish football. We have some good players and some real potential.”