What can you tell us about Eli Branic? Age? Alternate spellings of name? Civil War Soldiers & Sailors shows his name as "Branie".
If he was a citizen of the Cherokee Nation, it is likely he would be on either the Old Settlers Roll (Cherokee who moved west before the Treaty of New Echota 1835) or the Drennen Roll (Cherokee who moved west after the Treaty of New Echota). You can probably find both of these rolls at any library in Oklahoma and better libraries elsewhere. I don't think the Drennen Roll is online. It may be easier to find names like this in a book because of the many possible variations in spelling. You should also look at the 1880 Cherokee Nation Census and see if you find this surname. If you do not, I would assume he was not a citizen of the Cherokee Nation.
Can you find him on an 1860 US Census? The US Census will tell whether or not he was "white" (I think white and colored were the two choices). It will also tell his birthplace and that of his parents.
I'll do some more looking when I get home. Let me know what you find out.