Manchester City keeper Hart was at fault for Iceland's winning goal, allowing a shot from Kolbeinn Sigthorsson to creep into the bottom corner despite getting his hand to the ball.
Scottish international Wilson, who coached at Arsenal and became a TV presenter after winning the league and Cup double with the London side in 1971, said England's performance was the worst he had ever seen.
But he told Reuters: "I'm quite disappointed and upset for Joe because he's taking a lot of stick because it was his second high-profile mistake (of the tournament). I'm a big Joe fan. He will be fine. I don't think there is a technical problem."
Hart, the only player other than captain Wayne Rooney who spoke to the media after the defeat, was also criticised for allowing a Gareth Bale free kick to beat him in the 2-1 victory over Wales in the group phase.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
But speaking at the Wimbledon tennis championships where he was watching family friend Katie Swann in action, Wilson said the 29-year-old Hart, who played in two Manchester City Premier League title campaigns, should not be hounded.
"People were saying he was too tense, but Joe wears his heart on his sleeve. Some players are like that. Andy Murray plays his tennis like that, so it's wrong to say that he was too uptight. That is ridiculous," he said.
"Go back and look at all the internationals he's played and look at the Manchester City title-winning sides and look at how many mistakes he's made. In any one season the best goalkeepers in the world make mistakes. But look at how few they make."
England, who were due to travel home on Tuesday, have faced fierce criticism from the British media and former players. Wilson said the players were responsible for the embarrassing defeat.
"I think they have really let (coach) Roy Hodgson down," he said. "He has fallen on his sword because he's a very nice guy. But there was no game plan or anything.
"If I'm being brutal about it, I think it's about how much they care in the modern game. The financial side of the game now is almost to the point where young players are coming in, like Dele Alli, and are millionaires after one season."
Wilson echoed the widely-held view that the England squad lacked personality against a well-drilled Iceland team.
"There was no togetherness. They needed a John Terry type character to say 'look guys pull your fingers out'. "I don't think (captain) Wayne Rooney is a natural leader."
But he hoped the future would be fine for Hart, adding that goalkeepers had a difficult role to play.
"Go back to the Arsenal 1971 double-winning side," he said. "It sounds big-headed but I had an amazing season and everything came off for me.
"But who made the error that put us behind in the FA Cup final? I did. The rest of my life might have been Bob Wilson cost Arsenal the double. That's the goalkeeper's lot."
(Reporting by Martyn Herman. Editing by Adrian Warner.)
