I think the argument may be more that people will be less cautious about avoiding getting pregnant. They probably would. But that, for me, raises a need to address how people see the act of disposing of a potential baby. The education and values of people would be the problem, not the availability to the recourse of abortion. While we are about it, we can discharge the value placed on sex and the hold of religious attitudes used as social constraints because we no longer would have such a problem with unwanted babies coming to full term. By removing the need for the censure, we can disempower those that use that stick to enforce their version of hierarchy via professions of 'morality'. The main responsibility would be to avoid suffering of individuals who had to endure the choice to kill a fetus. Hence education, not rules.
The values of the most of people who adhere to religions are based in fear and a denial of responsibility, either out of fear of their own self needing to carry the burden, or of a fear towards allowing others the freedom of choice. To say, it is all God, and I trust it to him, and you must too, is just a type of mental and moral cowardice. The only defence of it is that we would have no morality without religion, and this is disproved. Still debated, but nonsense, in my eyes. How can you argue that we didn't have morality before God came along? We wouldn't have survived. To argue that God gave it to us is the same as saying it is inherent, and makes the religion redundant. Both branches of that argument fall flat.