MCMINNVILLE, Tenn.—Whitney Brown was in labor with her first baby when suddenly she couldn’t breathe.

Convulsions shook her body. Ms. Brown’s blood pressure and oxygen levels dropped, and the baby’s heart rate plunged. Nurses at Saint Thomas River Park Hospital called obstetrician Dawnmarie Riley, who minutes later burst into the operating room in such a rush her hospital scrubs were inside out.

Dr....