Various riders came down in the crash, including Bardet's teammate Tony Gallopin, who injured his shoulder, CCC Team's Simon Gesche – who was left with a broken collarbone less than six weeks after having broken his elbow at the Vuelta Ciclista a la Region Murcia – and Movistar's Marc Soler, who left the race with a fractured wrist.
The crash happened on a descent with six circuits left of stage 7 when the riders were said to be travelling at around 70kph, according to Gallopin.
"It happened very quickly, and I found myself thrown to the pavement and flew over my bike. Considering the violence of the fall, it could have been more serious," he said on the team's website.
"The crash was very severe because we were descending very fast," added Bardet. "I've hurt my ribs, but the effects could have been much worse. I am now waiting for the results of the additional examinations. This was not the expected result for this big block of races in March."
AG2r team doctor Arthur Molique confirmed the initial injuries of the two riders.
"Tony has suffered a trauma to his left shoulder, and will have additional examinations at the beginning of the week," said Dr Molique. "Romain has suffered a blow to the ribs with suspected fractures that will require further examinations."
Bardet was not set to race the Giro d'Italia in May, and will instead concentrate on the Tour de France, but his participation in the upcoming Ardennes Classics could be in jeopardy, depending on the outcome of his scans.
Thirty-three-year-old Geschke, of CCC, had only returned to racing at the Volta a Catalunya last week after breaking his elbow on the opening stage of the Vuelta Murcia in February.
Confirming the news of a broken collarbone on Twitter on Sunday evening, Geschke posted the x-ray of his injury, accompanied by the caption: "Really don't know what's going on this season, but so far it feels like it's cursed. Devastated."
The German's CCC teammate, Lukasz Owsian, also went down in the crash, but came away relatively unharmed.
"Both CCC Team riders went down at high speed, and while Lukasz escaped with only some abrasions, Simon was immediately taken to hospital for observation," explained CCC Team doctor Daniele Zaccaria.
"X-Rays revealed that he has sustained a broken right collarbone and he will require surgery at home in Germany. We expect the procedure to take place next week, and Simon should be able to resume training indoors five days after surgery with an expected return to racing after another five weeks."
Soler, meanwhile, went down with Movistar teammate Richard Carapaz, and while the later could remount and finish the race, scans of Soler's wrist diagnosed the Spaniard with a "non-displaced fracture to his scaphoid", according to a team press release.