I suppose we could seek to determine what sort of effect this will have on the abortion rate over the years to come. But, why bother? "They" would just shout it down anyway.
Why would you say such a thing? The number of women using contraceptives hasn't jumped from 14 to 67 percent. The number of women getting them for no copay has increased. Please tell me you're not one of those people who really believes that women who can't afford the 20 dollars a month for their pills would typically risk the expense of a costly abortion or the expense of raising a child by having sex with no protection against pregnancy. Women are a bit sharper than that, you know.
As far as what "they" might say... "They" might just mention that the abortion rate, along with the divorce rate and rate of sexual violence against women has increased substantially in the decades that hormonal contraception has become more and more widely used. While you conjecture about what this mandate might mean for abortion numbers, "they" might conjecture about how use of hormonal contraceptives impacts men and women's health, and the fact that the ACA offers no protections against expense for women who will suffer side affects and/or ensuing life threatening conditions from the meds Obama felt were so crucial to women's health.
Meanwhile, women who continue to struggle to pay copays for things like diabetes meds, anti-depressants, asthma meds, blood pressure meds, etc...aren't really feeling the love from the ACA. Obama basically took American women on a cheap date, and some are easily enough impressed.