The 32-year-old Enke was hit by a train travelling at 160 kph as it passed through a crossing near Hanover on its route between Hamburg and Bremen on Tuesday.
Enke is survived by his wife Teresa and an adopted eight-month-old daughter.
The German soccer federation (DFB) said Saturday's friendly with Chile had been scrapped because the players were in shock. All the players and coaches are due to attend a memorial service in Hanover in the coming days.
Enke's wife said the couple had tried to overcome years of depression through therapy.
"When he was acutely depressive it was a very difficult time because he lacked motivation and any hope of improvement," she told reporters at a news conference.
His wife, dressed in black and struggling to hold back tears, said: "It is crazy because now it is coming out anyway. We thought we could do everything and we could do it with love but you can't always do it."
"It was clear to everyone that we could not play," said DFB chief Theo Zwanziger. "We need time to process all this," he told reporters after a minute's silence before the news conference.
Germany team manager Oliver Bierhoff broke down in tears after adding that Enke's depression had gone unnoticed by team mates and officials.
German Chancellor Angela Merkel contacted Enke's family.
"The Chancellor communicated her shock and compassion to the widow of Robert Enke in a very personal letter," government spokesman Christoph Steegmans told reporters.
His doctor had treated Enke since 2003, during a turbulent time when the keeper had several unsuccessful transfers to clubs in Spain and Turkey. "He suffered from depression and fear of failure," Valentin Markser told reporters.
He lived in the shadow of Oliver Kahn and Jens Lehmann for almost a decade and it was only after the latter's international retirement last year that Enke emerged as a leading contender for the position, despite earning his first call-up in 1999.
