The 29-year-old Siutsou was involved in a pile-up at the back of a bunch of riders 50-km from the end of the 197-km third stage and was unable to make it back on his bike.
Another pile-up on the tight country roads of northern France also led to the withdrawal of Movistar's Jose Joaquin Rojas, second in the points classification in last year's race.
Wiggins, bidding to become the first Briton to win cycling's showcase event, is relying on his fellow Sky riders to help him in the punishing three-week event but they will now have to continue with just eight men.
Asked about the importance of Siutsou's withdrawal, Sky rider Bernhard Eisel told reporters: "He was an important part of the team. I'm sorry for him but we'll see if we can win the Tour with eight."
Wiggins's main rival, defending champion Cadel Evans, was buoyed by Team Sky's woes.
"You need to be firing on all cylinders in this race. To be one man down is not an advantage that's for sure," the Australian told reporters as he sat on the steps of the BMC team bus.
Wiggins trails overall leader Fabian Cancellara by seven seconds while Evans is 17 seconds behind the Swiss.