Joyce,
In answer to your question about Gayoso Hospital. Gayoso was a hotel in Memphis that was converted to a military hospital during the war. In answer to you question about being buried at Nashville. Yes that is possible. This is or was a U S Military Cemetary for Union Troops during the war so it is possible that one of the Williams is your relative.
Here is a blurb from one of the troops during the war concerning that hospital.

"Letter and envelope from William Taylor at General Hospital No. 3 in LaGrange, Tenn., to his sister Susan in Morrow County, Ohio, 1863 February 24

Scope Note:
Taylor discusses his illnesses and requests more letters from home. He notes that his regiment has left for Vicksburg, saying "I dont [sic] know whether I will ever get with the company a gain or not I can do more good here then [sic] I can there and the place where I can do the most good is the place for me." He also discusses his hope for a quick end to the war. He says "the time want [sic] be long till the union is restored as it wance [sic] was but wheather [sic] that will be or not I am able to say."

Folder 1 Item 2 Letter and envelope from William Taylor at Gayoso Hospital in Memphis, Tenn., to his sister Susan in Morrow County, Ohio, 1863 May 26

Scope Note:
Taylor says that he was sent out to fight, but that he was quickly reassigned to the hospital as a nurse. He notes that "the hospitals are all fixed up to receive the sick and wounded from Vicksburg wich [sic] they are looking for every day now." He also describes a visit to the camp site of his regiment.

Folder 1 Item 3 Letter and envelope from William Taylor at Gayoso Hospital in Memphis, Tenn., to his sister Susan in Morrow County, Ohio, 1863 December 17"

He apparently was at two hospitals as the letter discloses.
FC