Morgan was born in Goethenburg, the son of a clergyman, and grew up in North Wales. He studied classics at Oxford University, and originally intended to enter the church. However, after taking an interest in medical problems following a visit to the Scottish Highlands, he read medicine at St Mary's Hospital, London, qualifying in 1861. He took his MD in 1865 and became FRCP in 1868. He moved to Manchester after qualification, and became a physician at Salford Royal Hospital. He also lectured at Manchester Royal School of Medicine.
In Manchester Morgan was much concerned at the state of public health, and became secretary of the main public health pressure group, the Manchester and Salford Sanitary Association. He also helped establish the Manchester Nurses Training Institute, of which he became president. Morgan published a number of papers on public health subjects.
In 1867, he was made a physician at MRI, and in 1873 became first professor of medicine at Owens College Manchester. Morgan was a strong advocate of independent university status for Owens, particularly as a means of improving medical education in the north of England. He pushed strongly for the new federal Victoria University to have the right to grant medical degrees, which was achieved in 1883.