Majka's victory was the first on this Tour by his Tinkoff-Saxo Bank team, which lost main leader Alberto Contador when he crashed out injured on Stage 10.
Majka was not a threat to Nibali: He began the day 97 minutes behind the race leader, who has carried out a methodical, chipping-away strategy against his biggest challengers for the yellow jersey.
"I am really very happy," Majka, who was sixth in the Giro d'Italia this year, said of his first professional victory after chucking his stage winner's bouquet to the crowd. He became only the second Polish rider to win a Tour stage, after Zenon Jaskula in 1993. "I am a little tired, but … I had a calm first week to help Alberto. It broke my heart to see him leave."
Team owner Oleg Tinkov, a Russian businessman, choked up, wiped his nose, and put on sunglasses.
"We lost Alberto, we had to win," he said through a translator on French TV. "Rafal is a marvelous young rider. We will come back to try to win the Tour one day."
Majka said he did not believe speculation that Tinkoff-Saxo Bank selected him among its nine Tour riders only after the team suspended Czech rider Roman Kreuziger. Just days before announcing its Tour roster last month, the team suspended Kreuziger from competing any more this year because of anomalies in his biological passport, which cycling officials use to fight doping.
He has denied any wrongdoing.
Stage 15 preview
O Sunday, 6 a.m. TV • NBCSN
About the stage • Sunday's stage is a mostly flat but lengthy 138 miles across the lavender fields of Provence from Tallard to Nimes in southern France. The weather report is for storms and wind, which could add danger to a likely bunch sprint finish.
Leaders • 1. Vincenzo Nibali, Italy, 61 Hours, 52 Minutes, 54 Seconds. 2. Alejandro Valverde, Spain, 4 Minutes, 37 Seconds Behind. 3. Romain Bardet, France, 4:50. Also • 5. Tejay Van Garderen, United States, 5:49.