The 35-year-old, who won a silver medal at the 2004 Games, was involved in a collision along with a teammate last Friday.
Henze was travelling in a taxi that collided into a concrete barrier, causing serious head injuries which he failed to recover from.
A statement released by the German Olympic Committee read: "We are very sad today. Words can not describe approximately what we feel in the Olympic team after this terrible loss."
Team chief Michael Vesper added: "Today sports, for which the entire team came to Rio, moves into the background. Our thoughts are with his family who had the chance here to bid farewell."
A memorial service is due to be held in the Olympic Village on Tuesday, while the International Olympic Committee has been asked to fly the flag of Germany at half-mast at all Rio 2016 venues.