For me, the problem with the argument is that people get tied up in whether or not abortion is right/wrong or should be legal/illegal, rather than looking at what creates the prevalence of abortions.
According to the Guttmacher Institute, an organization that collects data around abortion rates and similar issues, the real lynchpin around this issue is the prevalence of unintended pregnancy, not whether or not abortion is legal.
So, the real action shouldn't be focused on whether or not abortion is legal -- that's secondary -- but on making certain that people have access to birth control, comprehensive sex education, and a culture that supports the use of birth control methods.
Most countries who have the lowest unintended pregnancy rates and therefore lowest abortion rates are countries whose national education systems provide comprehensive sex education as well as have national health care programs that provide free birth control to those who would seek it. As these two elements impact culture a great deal, a culture develops around utilizing these things and avoiding pregnancy, and thereby avoiding abortion.
Anyone who would want to prevent abortions would therefore want to educate the young populace about how to prevent unintended pregnancy. The problem is that they tend to focus on abstinence education, not wanting to discuss birth control, and so on. And in particular, not wanting to provide birth control.
Arguably, that needn't be done by "big gubment" (has anyone noticed that the government is about the same size regardless of whether the president is democrat or republican? i think if you *really* want small government, you're going to have to look at libertarians!), but part of the issue is that much of the anti-abortion-laws crowd is also focused on actively preventing birth control access and comprehensive sex education.
Which means, even if they succeed at making abortion illegal, the rates of abortion will be just as high (if not higher) than they currently are.