The cross-national correlation between contraceptive use and fertility is very large; it's -.66 for 172 countries (UN and CIA data). Not surprisingly, low fertility nations have high use rates. Rates in Europe are typically 60-plus percent for women of childbearing age who are in a relationship. The region with the lowest level is Western Africa where the numbers are roughly 10-20%.
Birth control is much more predictive of small families than abortion is. For 59 countries, the abortion-fertility correlation is only -.14. You might suspect that contraceptive use is inversely related to abortion and thus weakens the observed abortion-fertility correlation, but the fact is that high birth control countries also tend to be high abortion countries: the correlation is .12 for 59 countries. Reliance on birth control doesn't reduce abortion rates. Evidently, societies vary in their concern over fertility, and those with a high level of concern tend to focus on birth control as the first line of defense and abortion as a complementary backup.