What did prevent abortions was increased contraceptive use. The past decade has seen an increase in contraceptive use, a decrease in unintended pregnancies, and a decrease in abortions. The worldwide unintended pregnancy rate declined from 69 per 1000 in 1995 to 55 per 1000 in 2008. The proportion of married women using contraception increased from 54% in 1990 to 63% in 2003. It’s a correlation, not a definite causality, but I don’t think it’s a big jump to argue that increased contraceptive use led to decreased unintended pregnancies.
It seems clear to me that we know what to do to reduce the overall abortion rate and the terrible death toll of unsafe abortion: make contraction available to couples who want it.