Pharmacies are getting caught in the crossfire between religious liberty and patient rights.

There recently has been an upswing in reports of pharmacists refusing to fill prescriptions for birth control and emergency contraception because they object to the products on religious grounds. Patients denied necessary drugs say the conduct is discriminatory.

“If I were a pharmacy, I’d be nervous,” Sonia M. Suter, a professor at George Washington University Law School, told Bloomberg Law. Suter focuses on legal issues involving medicine, genetics, and bioethics.

Chain pharmacies, like CVS Health, Walgreen Co., and Wal-Mart Stores Inc., have adopted policies that try to...