The Narragansett Tribe referred to is not the actual Indian tribe but a part of a fraternal orgranization known as "The Improved Order of Red Men". Originally, this organization consisted of only white members. Ironically, Indians were not allowed to belong. Since your ancestor belonged to this organization in the time period he did, he is definitely not Indian.
I do not know what the I.O.R.M.'s rules for membership are today. You can learn more about them here:http://redmen.org/redmen/info/