Some hospital patients are heading home with a sheaf of prescriptions—and a bag of spinach and spaghetti squash.

Invoking the mantra that food is medicine, hospitals across the country are taking measures to prevent and treat illness through diet. To nudge patients into eating well at home, they have opened food pantries that offer nutrition counseling and healthful fare. They are growing their own produce, adding farmers to the payroll and hosting greenmarkets. A few are even tiptoeing into the grocery business.