By mid-war, each regiment usually had one, more senior, surgeon (with the equivalent rank of major) and one, more junior, assistant surgeon (with the rank of captain), although early in the war two assistant surgeons might be assigned, especially in Union regiments. The most senior surgeons could be detached from their regiment and placed in charge of a brigade, division or corps hospital(s). Even an assistant surgeon would have a medical degree, or close to it. Something equivalent to a medic was the regiment's Hospital Steward. Even a few privates might be pressed into service as nurses in regiments that sustained mass casualties after a large battle.
Some of the better surgeons augmented their medical degrees with additional training in European medical schools, which were widely believed to be at the cutting edge of the medical profession. A handful had even served in a war zone. I know of 24 medical colleges or schools, north and south, that had turned out surgeons and assistant surgeons for both sides.
A few examples:
(CSA) Surgeon John Thompson Darby, Maj. Gen. Hood's staff. Attended Medical College of South Carolina; obtained M.D. from University of Pennsylvania, Medical Department, 1858; graduated from Jefferson Medical College with honors.
(CSA) Assistant Surgeon Alexander Rives, Jr., 15th Alabama. Attended University of Virginia, Medical School, 1857-1859; graduated from New York University, Medical College in 1861.
(CSA) Surgeon Erwin James Eldridge, Cobb's Georgia Legion. Jefferson Medical College, 1854, M.D.; studied in Vienna; served in the Crimean War.
(CSA) Brigade Surgeon George Rogers Clark Todd (brother of Mary Todd Lincoln). Transylvania University, Medical Department, M.D., 1850.
(CSA) Surgeon William Riddick Whitehead, 44th Virginia. VMI graduate, attended University of Virginia, Medical School, 1851-1852; obtained M.D. from University of Pennsylvania, Medical Department, 1853; M.D. from School of Medicine, Paris, France, 1860; Professor of Clinical Medicine, New York Medical College, 1860-1861.
(CSA) Assistant Surgeon Samuel Taylor Holliday, 27th Virginia. Obtained M.D. from University of Pennsylvania, Medical Department, 1860.
(CSA) Surgeon-in-Charge of Third Corps hospitals, Henry DeSaussure Fraser. Medical College of South Carolina, 1851, M.D.; afterwards worked at a hospital in Paris.
(USA) Assistant Surgeon Edgar Parker, 13th Massachusetts. Harvard College, Medical School, 1863, M.D.
(USA) Surgeon Nelson Isham, 97th New York. Graduated 1828 from Yale, School of Medicine.
(USA) Assistant Surgeon Richard Powell, 88th New York. Graduated from Licentiate Royal College of Surgeons, Ireland.
(USA) Assistant Surgeon John Ryan, 9th Massachusetts. Harvard College, Medical School, 1860, M.D.
(USA) Division Surgeon Daniel Garrison Brinton, Fifth Corps. Graduated from Jefferson Medical College in 1860; student in Paris and Heidelberg, 1861.
(USA) Division Surgeon Elias W. H. Beck, 19th Indiana. Attended Ohio Medical College; New York University, Medical College, 1848, M.D.