On July 6, Bill McDermott, the chief executive of one of the world's largest software companies, fell down the stairs at his brother's house. He awoke in a pool of blood. Mr. McDermott was holding a drinking glass when he fell, and it shattered and entered his face, leaving a vertical scar from above his left eye to below his ear. The glass also entered his left eye, causing him to lose the organ after unsuccessfully undergoing several surgeries and procedures to keep it. The experience is spurring Mr. McDermott, 54, to move his company, SAP — a firm based in Germany that specializes in large-scale software for manufacturers, retailers and the like — deeper into health care.