The Slovak beat Norway's Alexander Kristoff and Frenchman Arnaud Demare, who were second and third, respectively.
Team Sky's Froome was struck by at least one spectator a little under seven kilometres from the summit of the famous climb and seemed to be spat at shortly after, on his way to a fourth-placed finish as team-mate Geraint Thomas took victory.
On stage 14 to Mende on Saturday, the three riders at the top of the overall classification all matched each other on the final climb. Thomas will want his first grand tour win while Froome aims for history with a record-equalling fifth title. The trio lost a handful of seconds to Primoz Roglic (LottoNL-Jumbo), but also put time into the likes of Movistar pair Mikel Landa and Nairo Quintana, and AG2R La Mondiale's Romain Bardet.
Facing no attacks on the flat stage, Thomas had no trouble maintaining his advantage of 1 minute, 39 seconds over defending champion Froome.
The action came a day after Chris Froome was the target of slaps and shoves on his way up Alpe d'Huez and just hours after Christian Prudhomme, the race director, renewed his calls for roadside fans to behave themselves.
As for the jeers directed toward him and Froome by fans skeptical of Froome's clearance from doping allegations days before the start of the Tour, he said it was part of being in the spotlight.
Nibali sustained a fractured vertebra when he crashed about four kilometres from the top of the 13.8-km climb after being hit by what appeared to be either a spectator or a police motorbike. "Froome closed the gap and then attacked, and it was Thomas who closed the gap to Froome".
"I didn't actually notice it", Thomas said.
Thomas has said that although he would work to support Froome, he is not going to intentionally lose time. "Same for Froomey. The most important thing is the group is strong and we have the best position". "It was nice to go on fast roads, the wind was kind to us and it was a good day to get ticked off".