1870 census for Marshfield, Washington, VT family #27:
Heath, Nathn L., 38, b. Vermont
Heath, Jane(?), 28, b. Vermont
Heath, Jennie M., 7, b. Vermont
Heath, Chas., 5, b. Vermont

1875 MN census for Brainerd, family #96:
Heath, N.L., 42, b. Vermont
Heath, J.M., 34, b. Vermont
Vassor, J.M., 13, b. Vermont
Heath, Hellen, 3, b. Vermont
Heath, Charles, 9, b. Vermont
Heath, Harriet, 0, b. Minnesota
(J.M., 13, is Josephine M., step-daughter of Nathan. Mary J Vassar marries Ole Evenson in Crow Wing county on 12/06/1880.)

1880 census for Brainerd, Crow Wing, Minnesota:
Heath, Nathaniel, 50, b. Vermont, carpenter
Heath, Josephine, 39, b. Vermont
Heath, Charles H., 14, b. Vermont
Heath, Hellen, 9, b. Vermont
Heath, Harriet, 5, b. Minnesota
Heath, Adelia, 3, b. Minnesota

Index to Pension files has Nathan L. (not G.) Heath. So does the census.

Veteran's Headstone contract for Nathan G.(sic) Heath (see image) has 06/30/1883 date of death.

Crow Wing County Death Records, Book A, Page 8, Line 20: Nathen L. Heath, 52y 5m 19d, married, died Jany. 29, 1883 in Brainerd of Lung Fever. Occupation: Laborer. Arrived in Minnesota October 1873. Born in Vermont to Jerry & Sarah Heath, both born Mass.

So there is a death date conflict and a middle initial conflict between the county death record and the headstone contract. I'm going with the county death record for both.

1883-01-29 minus 52-05-19 = 1830-08-10 birth date.

1870 census for Marshfield, Washington, VT family #27:
Heath, Nathn L., 38, b. Vermont
Heath, Jane(?), 28, b. Vermont
Heath, Jennie M., 7, b. Vermont
Heath, Chas., 5, b. Vermont

1875 MN census for Brainerd, family #96:
Heath, N.L., 42, b. Vermont
Heath, J.M., 34, b. Vermont
Vassor, J.M., 13, b. Vermont
Heath, Hellen, 3, b. Vermont
Heath, Charles, 9, b. Vermont
Heath, Harriet, 0, b. Minnesota
(J.M., 13, is Josephine M., step-daughter of Nathan. Mary J Vassar marries Ole Evenson in Crow Wing county on 12/06/1880.)

1880 census for Brainerd, Crow Wing, Minnesota:
Heath, Nathaniel, 50, b. Vermont, carpenter
Heath, Josephine, 39, b. Vermont
Heath, Charles H., 14, b. Vermont
Heath, Hellen, 9, b. Vermont
Heath, Harriet, 5, b. Minnesota
Heath, Adelia, 3, b. Minnesota

Index to Pension files has Nathan L. (not G.) Heath. So does the census.

Veteran's Headstone contract for Nathan G.(sic) Heath (see image) has 06/30/1883 date of death.

Crow Wing County Death Records, Book A, Page 8, Line 20: Nathen L. Heath, 52y 5m 19d, married, died Jany. 29, 1883 in Brainerd of Lung Fever. Occupation: Laborer. Arrived in Minnesota October 1873. Born in Vermont to Jerry & Sarah Heath, both born Mass.

So there is a death date conflict and a middle initial conflict between the county death record and the headstone contract. I'm going with the county death record for both.

1883-01-29 minus 52-05-19 = 1830-08-10 birth date.