Henze was travelling with a teammate in a taxi to the athletes' village early in the Barra de Tijuca neighbourhood when the car crashed into a concrete barrier. His teammate Christian Kaeding suffered minor injuries and was briefly treated in hospital before being released.
It was not yet clear how the accident occurred.
"We are endlessly sad on this day," said German Olympic Sports Confederation (DOSB) President Alfons Hoermann in a statement. "Words cannot come close to describing how we in the Olympic team feel after this dreadful incident."
The 35-year-old, a 2004 Olympic Games silver medallist, had undergone emergency surgery early on Friday and had been in intensive care ever since.
"Today sports, for which the entire team came to Rio, moves into the background," said team chief Michael Vesper. "Our thoughts are with his family who had the chance here to bid farewell."
The German team said it will hold a ceremony on Tuesday (local time) at the Place of Mourning inside the Olympic Village where the athletes live and has asked the International Olympic Committee to fly all German flags in the venues at half staff.