If everyone in Israel did teshuva [became Torah observant], then how would hospitals be run on Shabbat?
Better than they are now. You see, the Torah views the saving of a life as a mitzvah of paramount importance. This is seen reflected in emergency health care services run by Orthodox Jews, which have a faster average response time than others. In fact, there are already hospitals in Israel that are run according to Jewish law - for example, Shaarei Tzedek Hospital in Jerusalem and Laniado Hospital in Netanya.
The Torah commands us to break Shabbat in order to save a life. Therefore, there is no contradiction between working in a hospital on Shabbat and observing Jewish law. Actions that do not pertain to saving life are done after Shabbat. (For example, filling out forms.)