If you look at the Official Army Register, available for example at HathiTrust, you’ll see the regimental officers include surgeons and assistant surgeons.
The law authorizing the volunteer army (United States Statutes at Large 12 (1859-1863), chapter 9, pp.268-271; on the Library of Congress webpage) says:
‘SEC. 2. And be it further enacted, That the said volunteers shall be subject to the rules and regulations governing the army of the United States, and that they shall be formed, by the President, into regiments of infantry, with the exception of such numbers for cavalry and artillery, as he may direct, not to exceed the proportion of one company of each of those arms to every regiment of infantry, and to be organized as in the regular service. Each regiment of infantry shall have one colonel, one lieutenant-colonel, one major, one adjutant, (a lieutenant,) one quarter-master, (a lieutenant,) one surgeon and one assistant surgeon, one sergeant-major, one regimental quartermaster-sergeant, one regimental commissary-sergeant, one hospital steward, two principal musicians, and twenty-four musicians for a band, and shall be composed of ten companies, each company to consist of one captain, one first lieutenant, one second lieutenant, one first sergeant, four sergeants, eight corporals, two musicians, one wagoner, and from sixty-four to eighty-two privates.’
‘Sec. 3. And be it further enacted, That these forces, when accepted as herein authorized, shall be organized into divisions of three or more brigades each; and each division shall have a major-general, three aides-de-camp, and one assistant adjutant-general with the rank of major. Each brigade shall be composed of four or more regiments and shall have one brigadier-general, two aides-de-camp, one assistant adjutant-general with the rank of captain, one surgeon, one assistant quartermaster, and one commissary of subsistence.’
The act of 2 July 1862 (Statutes at large, vol. 12, chap.127, p.502, available on the Library of Congress webpage) added 40 surgeons & 120 assistant surgeons of volunteers, and said:
‘SEC. 2. And be it further enacted, That from and after the passage of this act Brigade Surgeons shall be known and designated as Surgeons of Volunteers, and shall be attached to the general medical staff under the direction of the Surgeon General; and hereafter such appointments for the medical service of the army shall be appointed Surgeons of Volunteers.’
I haven’t looked for other amendments. But yes, privates weren’t surgeons.