Notes: "Various authorities, both ancient and modern (starting with Herodotus) have sought links with Egypt to explain the origins of certain medical practices, especially surgery in the Greek world. Others have found links with the near east, and with Babylonian medicine in particular, although these have proved very difficult to prove. Some argue that the Hellenistic doctors working in Alexandria continued to be influenced by Egyptian traditional medicine in the 4th and 3rd cents. BC. In the 5th cent. BC, when Herodotus told the story of Darius' Greek physician, Democedes of Croton, the really surprising feature of his career -- apart from its conspicuous success -- was Democedes' technical superiority over the Egyptian doctors." Source: Oxford Classical Dictionary (3rd ed. revised).
According to the Ancient History Encyclopedia (1), "the Egyptian Imhotep (2667 - 2648 BCE) is the first physician in history known by name. The earliest known surgery in Egypt was performed in Egypt around 2750 BCE. The Kahun Gynaecological Papyrus treats women's complaints, including problems with conception. Thirty four cases detailing diagnosis and treatment survive, some of them fragmentarily. Dating to 1800 BCE, it is the oldest surviving medical text of any kind. Medical institutions, referred to as Houses of Life are known to have been established in ancient Egypt as early as the 1st Dynasty."
(1) This quote is taken from an article which no longer exists. The Ancient History Encyclopedia apparently reoorganized their entries for ancient medicine, dividing the subject into various categories which can be found here.