CARACAS, Venezuela—His heart failing fast, Pedro Gonzalez checked into one of Venezuela’s leading public hospitals in September, hoping for a new heart valve to save his life. He prayed day and night for a successful surgery, clutching a blanket embroidered with an image of the Virgin Mary.

In late November, the director of cardiovascular surgery at the University Hospital sent out letters to the cardiology ward’s patients, telling them they were being discharged. The reason cited: a dearth of operating-room supplies—no catheters,...