The New Orleans Public Library an index, the Louisiana Biography and Obituary Index, which references obituaries and death notices published in New Orleans newspapers from 1804-1972 and biographical information published in older Louisiana collective biographies.

Captain George P. Ring is listed in this index as having an obituary published in the Daily Picayune newspaper as follows:

Ring, Capt. George P
Death date: 1867-06-09
Age: 37 years
Sex: M
Obituary citation(s): Daily Picayune, 1867-06-11, Pg. 4 col. 3

You can order a copy of this obituary from the library. I was thinking that the obituary may provide his place of burial. It's worth a try. Click on the following link for ordering information:

http://neworleanspubliclibrary.org/obits/orderingobits.htm

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George P. Ring is also listed in the New Orleans, Louisiana Death Record Index 1804-1949 available at Ancestry.com as follows:

Name: George P. Ring
Age: 37 Yrs
Death Date: 9 Jun 1867
Color: W
Page: 318
Volume: 40

A copy of the death record can be ordered from the Louisiana State Archives in Baton Rouge.

http://www.sos.louisiana.gov/tabid/88/Default.aspx

Regards----