The demise of St. Vincent's Hospital in Greenwich Village two years ago has led to a struggle for health care supremacy in some of New York's most distinctive neighborhoods, offering a glimpse, in the process, at what might be the future of urban medicine. Without building a hospital, one large chain, Continuum Health Partners, is establishing a beachhead by connecting with outpatient clinics, trying to dominate the market and create a feeder network for its hospitals in other neighborhoods. It is joining forces not just with traditional clinics but also with newer experiments like doctors working out of drugstores. A competitor, NYU Langone Medical Center, is expanding its physician practices downtown, and like Continuum, it has hired dozens of stranded St. Vincent's doctors.