Are you sure?
100% of riders, for all rides they take, would choose to use a car if they could not use ride sharing?
Absolutely none would walk, ride a bike, take a bus, take a train, choose not to go?
I have a hard time believing that it's purely a "car I have access to or a ride share," 100% of the time. So I believe some amount of rides are happening that would not have people driving.
Of the drivers for ride sharing, there is some amount of time between trips that they are simply waiting for the next customer. This is time they wouldn't be on the road if they were not driving, and time the person who is using the ride sharing wouldn't be on the road either. So, I also believe that drivers must be on the road more often than the sum of the time they've kept other people from driving.
If drivers are on the road more with ride sharing, and if less than 100% of people using the ride sharing service would choose a method of transport other than a car, I cannot see how ride sharing wouldn't strictly require an increase in traffic?