De Gea's poor punch led to Clint Dempsey's equaliser at White Hart Lane on Sunday as Tottenham denied United three points.
It was the latest in a long line of mistakes from De Gea, 22, since his £19m move from Atletico Madrid in 2011.
Carroll told the Daily Star: "Once you make mistakes, people never forget about it, so I think as a young goalkeeper it's very hard.
"Manchester United are one of the biggest teams in the world and if you make mistakes the press are always on your case all the time. That's what happens at a big club."
Olympiakos keeper Carroll added: "I feel very sorry for him. I had four years there but I made a lot of mistakes as well.
"You need the manager's backing and I think he's got that because he's played a lot of games since he's come in.
"He's a very good keeper. He's still young.
"There have been a lot of changes at Manchester United across the back four this season with injuries, so I think it's very difficult for him at the moment."