Researching the family legend of a native American ancestor I found a female of unknown surname but unusual given name (Trasda) and inconsistent census race - usually counted as white, but occassionally indian. But given the timeline - marriage about 1828 in Tennessee, such a marriage would be illegal under Tennessee law as with many states. Perhaps this explains the lack of marriage certificate.
But given how common the story of native admixture is, there has to be many such marriages. How enforced were miscegenation laws for I/W couples? Is this something that went on in the most rural areas only? I know there was plenty of "marriages" with natives eventually discarded for white wives to "settle down" but this case was a life long marriage, eventually moving to Texas right after it joined the union.
While I don't know the particular laws of Tennessee in the era concerned, there are couple of potential explanations for your ancestor's case, which touches on some interested historical themes, like the role of the state in regulating marriage and the fluidity of racial identity.
First, there might be no marriage certificate because of miscegenation laws, although from what I understand many of those laws originate later than you might imagine. Check out "Public Vows" by Nancy Cott.
I get the distinct impression that it was easier, in some cases, to subsume being racially categorized as "Indian" for a woman marrying a white man, than it would have been if their races had been reversed. And generally speaking, a lot of interracial marriages and families in the South in that era with Native Americans got swallowed up by black/white racial categorization, sometimes with bizarre and painful long-term effects. Newt Knight and the "Free State of Jones" story is a good example - see Victoria Bynum's book on that, "The Free State of Jones: America's Longest Civil War" (UNC Press, 2003), not the Sally Jenkins rip-off. For a good study on the interplay of race and Indian identity, see Claudio Saunt, "Black, White, and Indian," (OUP: 2005).