Senators sent the bill to the House with a 31-5 vote Monday.
But the prohibition only would take effect if a federal appeals court upholds a similar law in Mississippi.
Milkovich says he is trying to protect unborn babies.
Opponents say the ban would eliminate abortion as an option before many women even realize they are pregnant, in violation of the U.S. Supreme Court’s Roe v. Wade decision that legalized abortion.
Several states have enacted similar bans this year.