The law requires doctors who perform abortions to have staff privileges at a local hospital.
The ACLU said some providers are not able to obtain those privileges because the local hospital opposes abortions or the doctors live too far from the hospitals.
The law would force three of the state's five abortion clinics to stop providing the procedure, the ACLU said.
The Planned Parenthood Federation of America and Planned Parenthood Southeast joined the lawsuit.
"This law is part of a coordinated national campaign designed to outlaw abortion, state by state," Alexa Kolbi-Molinas, staff attorney at the ACLU Reproductive Freedom Project, told The Hill.
"This law insults women's intelligence by claiming to be about safety, when the true intent is to shut down clinics and prevent a woman from making a real decision about her pregnancy."
Earlier this year, a federal court blocked a similar law in Mississippi.