Gov. John R. Kasich, a Republican who is running for president, opposes abortion but has not yet taken a position on this bill. Since his election in 2010, he has signed a variety of abortion restrictions, including a law requiring women to have an ultrasound and be offered a chance to see an image of the fetus before undergoing the procedure.

Mike Gonidakis, the president of Ohio Right to Life, said his group had made the bill here a legislative priority because Down syndrome is so recognizable, so easily diagnosed in pregnancy — and so likely to lead to abortion.

“We all want to be born perfect, but none of us are, and everyone has a right to live, perfect or not,” he said. “You go to any supermarket or mall and see these families who just happen to have a child with Down syndrome, and they will tell you how fortunate they are to have those children. Pretty soon, we’re going to find the gene for autism. Are we going to abort for that, too?”