Porte crashed on the descent from the Mont du Chat, the last of the day's big climbs, as he slid off the inside of corner and was violently flung across the road, with Quick Step Floors rider Dan Martin unable to avoid him.
However, organisers said he was conscious and speaking but was being taken to hospital.
At the time of the crash, AG2R La Mondiale were lighting up the race on the descent before the next hors categorie climb.
Tour de France doctor Florence said Porte had spoken after the fall but that the extent of his injuries would not be known until after X-rays.
In the confusion that followed, Bardet used his local knowledge of the roads to ride clear and catch his compatriot as the road levelled out in the run into Chambery.
"He (Porte) did it four times before the race, he did it in training, he did it in a race (the Criterium du Dauphine)". Dutchman Robert Gesink (LottoNL-Jumbo) and Italy's Manuele Mori (UAE Team Emirates) fell inside the opening 10km and both were forced out, the former with a fractured vertebrae less than a day after he finished runner-up in Stage 8.
Porte had been having an exceptional day on the bike, riding in the yellow jersey group chasing the last remaining breakaway rider, Warren Barguil (Sunweb), when he appeared to misjudge a bend.
One sprinter who will not be there to fight for a second win is Arnaud Demare after the French national champion - plagued by illness - was eliminated with three of his FDJ team-mates for finishing outside the time limit.