The MEASURE Demographic and Health Surveys project published an analytical study on Recent Trends in Abortion and Contraception in 12 Countries in 2005, the 12 countries in the study being Armenia, Azerbaijan, Georgia, Kazakhstan, the Kyrgyz Republic, Moldova, Romania, Russia, Turkey, Turkmenistan, Ukraine, and Uzbekistan. From the Executive Summary of the study:
In a series of simulation models, the implications for further reductions in the
prevalence of abortion are estimated. For example, if the women currently using
modern methods of contraception were joined by those currently using traditional
methods, abortion rates on average could be reduced by 23 percent; if women classified as having unmet need for family planning were also added to this group, abortion
rates could be reduced by as much as 55 percent.
The study found strong patterns of abortion declining with modern contraceptive prevalence rising in several countries:
In summary, 8 of the 12 countries show declines in abortion accompanied by increases in the prevalence of modern methods; one shows an increase in abortion with little change in prevalence (Azerbaijan); two show increases in abortion along with an increase in prevalence but with a substantial reduction in the ideal number of children
(Turkmenistan) or a very low ideal number of children with increasing contraceptive
failure rates (Ukraine); and one features no net change in abortion rates but with increasing contraceptive prevalence offset by increasing rates of contraceptive failure
Contraception in the Netherlands: the low abortion rate explained seems to strongly suggest that contraception is a major factor in the country's low abortion rates:
This article gives a review of the main factors that are related to the low abortion rate in the Netherlands. Attention is payed to figures on abortion and the use of contraceptive methods since the beginning of the 1960s up to the end of the 1980s. The strong acceptance of family planning was influenced by changing values regarding sexuality and the family, the transition from an agricultural to a modern industrial society, rapid economic growth, declining influence of the churches on daily life, introduction of modern mass media and the increased general educational level. The introduction of modern contraceptives (mainly the pill and contraceptive sterilization) was stimulated by a strong voluntary family planning movement, fear for overpopulation, a positive role of GPs, and the public health insurance system. A reduction of unwanted pregnancies has been accomplished through successful strategies for the prevention of teenage pregnancy (including sex education, open discussions on sexuality in mass media, educational campaigns and low barrier services) as well as through wide acceptance of sterilization. The Dutch experience with family planning shows the following characteristics: a strong wish to reduce reliance on abortion, ongoing sexual and contraceptive education related to the actual experiences of the target groups, and low barrier family planning services.
Unfortunately I don't have access to the full article.
The 1999 book From Abortion to Contraception: A Resource to Public Policies and Reproductive Behavior in Central and Eastern Europe from 1917 to the Present examines the "abortion culture" in USSR successor states, and theorizes that a key factor in high abortion rates was the limited availability, high cost and bad press of modern contraceptives.
Although I haven't got the slightest idea if freely available contraception will (significantly) cut the abortion rate in the US and I'm guessing that data from USSR successor states won't be enough to convince US conservatives, after giving a second read to the MEASURE DES study I'd say it's definitely worth a try.