The Muslims inherited and developed Unânî Medicine (Greek Medicine). We also acquired knowledge of medicine through The Family of the Prophet (Ahl Al Bayt) and the Shia but there is no conclusive proof that their medicine is directly from the Prophet.
Greek Medicine was translated towards the latter stages of the Umayyad (Umawi) Caliphate to the early stages of the Abbasid Caliphates (Abbasi Khalîfa)
There are many great Muslim scholars who were scholars of Greek Medicine but not prophetic medicine i.e. Farâbi, Bayrûni, Ibn Sîna
Initial Material on Prophetic Medicine (Tibb An Nabawi)
At Tibb An Nabawi – The first Book on this subject published on this subject. By Ali Ar Riddâh (a descent of the Prophet), 11th Imam of the Shi’a.