It would exist because there is anonymity in a free-society -- your medical and doctors records are just that YOURS.
So how would you find out -- you wouldn't even know how many were taking place.
The reason why the number of abortions would be FEWER is because the value of women and babies (as future consumers and entrepreneurs) would be much higher.
We (as a society) would have a culture wherein mothers and children were protected and cared for -- not forever in the welfarist sense, but just on face value.
*&^ Constitution --- Constitutional Rationality