"Jack's health is the team's top priority and he will continue to be monitored closely by the team's medical staff."
Sport director Charley Wegelius said it was a "stressful day even for the Tour de France".
"High winds in multiple direction changes made positioning in the peloton extremely difficult. It resulted in multiple crashes. We're very sad to lose Jack and we will be thinking of him as we continue the race."
Compatriot Greg Henderson, riding with Lotto Soudal, was 135th in the stage, 14 minutes behind teammate Andre Greipel who captured his second stage win of the tour.
Bauer was the second rider to abandon overnight following French sprinter Nacer Bouhanni (Cofidis) whi was ninth overall.
With crashes taking down riders on the rain-drenched roads, keeping team leaders safe was the order of the day.
By the time Greipel attacked in the last 100 metres to win, the last of the day's seven crashes had taken down 30 riders.
There had been a big spill in stage three, too, involving some 20 riders. But once again, Chris Froome and the other Tour contenders avoided them.
"There was absolutely everything out there today. It rained, which made the roads slippery, and it was also windy," said Ian Stannard, Froome's Team Sky teammate. "That made for a stressful day."
Peter Sagan, a Slovak rider seeking to win the green jersey as the Tour's best sprinter for a fourth straight year, had even more reason to feel tired.
He spent most of the day protecting his Tinkoff-Saxo teammate Alberto Contador, and then contested the stage sprint, zooming ahead of British rider Mark Cavendish to take second place behind Greipel.
"It was also very crazy today with rain, wind and a lot of crashes and I'm happy with how we finished," Sagan said. Three of his teammates were involved in crashes, but not Contador.
"Everybody wants to be at the front on a day like this to protect the team leaders and that creates tension," Sagan said. "I want to help and protect Alberto."
The yellow jersey group rolled over the line with no change to the leading positions.
German rider Tony Martin, the winner on stage 4, still leads Froome by 12 seconds and Tejay Van Garderen, a promising American rider with strong climbing skills, by 25.
"Everyone thought today was going to be the relaxed day of the tour. But the wind and the rain made it anything but," Van Garderen said. "Luckily, I have one of the strongest teams here."
Among the main contenders, Froome leads two-time Tour champion Contador by 36 seconds; defending champion Vincenzo Nibali by 1:38 and Colombian rider Nairo Quintana, the 2013 runner-up, by 1:56.
The stage took the weary peloton over 189.5 kilometres (117.5 miles) from Arras to Amiens in northern France, passing some of the battlefields of World War One.
The rain, which largely stayed away the day before, thundered down and turned the roads into something of an ice rink.
Clinching the eighth stage win of his Tour career on damp tarmac, Greipel punched the air in delight.
Stage 6 is another mostly flat stage for sprinters, taking the pack over 191.5 kilometres (119 miles) from Abbeville to Le Havre, France's biggest commercial port.
