Abortion rights activists are suing to block one Ohio abortion restriction while another, stricter bill has a chance to become law this session.

Ohio Planned Parenthood groups sued Feb. 14 to nullify a state law imposing civil and criminal penalties on doctors who perform a dilation-and-evacuation abortion procedure before stopping a fetus’ heartbeat. The suit comes as the legislature takes another crack at a bill that would bar abortion procedures altogether if a fetus has a detectable heartbeat, widely called a “heartbeat bill.”

The push and pull over abortion restrictions in Ohio mirrors similar developments across the country where GOP-controlled...