BANGALORE -- Hair tucked into a surgical cap, eyes hidden behind thick-framed magnifying glasses, Devi Shetty leans over the sawed open chest of an 11-year-old boy, using bright blue thread to sew an artificial aorta onto his stopped heart.

As Dr. Shetty pulls the thread tight with scissors, an assistant reads aloud a proposed agreement for him to build a new hospital in the Cayman Islands that would primarily serve Americans in search of lower-cost medical care. The agreement is inked a few days later, pending approval of...