Look under Health Information -- Health Statistics for links to a few collections of statistics on specific topics from the US National Institutes of Health.
World Health Organization data on country-level international health indicators. Tables, maps, data API, reports, and data query tools organized by broad topic. Scroll down for topics or click 'Indicators' to see all available measures.
The World Health Organization provides statistics on 70 core indicators for the 193 WHO member states, as well as extensive data on other international health concerns. Some indicators are are easier to find on this older platform. See also the newer Global Health Observatory.