The 66-year-old former professional tennis player believes the British number one lost during the round of 16 match against Alexander Bublik in the French Open on Monday (02.06.25) because he was "stressing".
Draper has previously admitted he was battling anxiety, but McEnroe said he needs to compose himself if he wants a shot at being a major champion in the future.
McEnroe told TNT Sports: "It looked like the nerves got to Jack.
"It looked like Jack was stressing. He has been working on that.
"To me, that is going to be a factor in whether he wins multiple majors or not, to be able to handle those emotions.
"He was playing well, to me, and I was surprised that he was that stressed out."
Draper - who leapfrogged the 24-time Grand Slam champion Novak Djokovic, 38, and took the fifth spot after he beat Matteo Arnaldi in their quarter-finals match of the Madrid Open - lost to Bulbik 5-7 6-3 6-2 6-4 on Court Suzanne Lenglen at Roland Garros.
And Bublik said beating the Brit was the "best moment" of his life.
He said after the match: "You know, sometimes in life, there is only one chance, I had a feeling that was mine.
"I couldn't let it slip, so standing here, it's the best moment of my life. Period.
"I'm standing here like I won the thing, but at the end, what can I say. It's amazing.
"I can't cry here, come on let me be in peace. I still have a match to go, I'm a professional tennis player, I have to get ready."
