The 18-year-old suffered a muscle bundle tear in his left thigh during training on Friday in Chicago and was taken to hospital. He will be replaced by Assan Ouedraogo.
"I feel incredibly sorry for Lenny," Nagelsmann said. "With his light‑heartedness, his creativity, his pace, and his personality, he fit into the team perfectly."
Karl enjoyed a meteoric rise this season as he helped Bayern clinch the domestic double and reach the Champions League semi-finals.
"It is a huge shock for him and for all of us that he will miss the World Cup. It is only a small consolation that he is young and still has many tournaments ahead of him," Nagelsmann said.
"With Assan Ouedraogo, we are now getting a player who, similar to Lenny, had a great start with us. He is also highly talented and is meant to play here boldly and freely."
Germany arrived in the United States on Tuesday for the World Cup starting next week.
The four-time champions have set their sights on winning the title after shock first-round exits at the previous two tournaments, in 2018 and 2022.
The Germans, who have not tasted international success since winning their last World Cup title in Brazil in 2014, are in Group E and kick off their campaign against Curacao on June 14th before facing the Ivory Coast and Ecuador.