There is a long-held story of Choctaw heritage, which led very nearly to being officially included in the Choctaw Nation in the 1990's. As this has not appeared in DNA samples of the younger (as of this writing, at least!) Stewarts, there has been some concern on the Stewart lineage as a whole, because surely there remains merit in the story of Choctaw heritage?
To summarily say it at the start: No. I have no evidence to support any amount of Native American heritage within the Stewart line, Choctaw or otherwise, and in fact I find only evidence to the contrary. Ultimately, I think it boils down to the desire to offset inherited guilt due to the horrific genocide of the Indigenous Peoples of America at the hand in European colonizers, which is not uncommon in many American families (and has cropped up in American politics as well, see Elizabeth Warren's familial controversy!)
That said, it did not come from nowhere; let's look at this from the top. According to tradition, Saphronia E., wife of Rufus Stewart, is where the Choctaw heritage comes in. When her granddaughter Margaret Azilee Stewart was born, Saphronia looked at her and said she was "another little Choctaw," leading to her nickname "Choc." In the 1990's, Saphronia was found on the Dawes Act list with a 3-digit roll number, listing her as a full Choctaw by blood. Thus, the swearing up and down that the Stewarts came from a Choctaw heritage is confirmed! Easy, right?
Well...
In the Dawes Act listing (page 6 of the Final Rolls of Citizens and Freedmen), created on September 25, 1902, there is a Sophronia E. Nail with the following information:
889, Sophronia E. Nail, age 12, Female, 3/4th Choctaw, Census Card No. 467
This same Sophronia has several additional documents, including letters about duplicate applications (Dawes Enrollment Jacket for Choctaw, Choctaw by Blood, Card #4890), in which she was originally added to the Dawes Act list in 1899 at age 9 with a Dawes Act Number of 423. When the duplicate was found, the first was excised and the second kept. These documents also include additional information; Sophronia was living in Tobucksy County, Choctaw Nation, which would become McAlester, Pittsburg County, Oklahoma. Her father was Joe Nail and her mother Vicey Steward (Enrollment for Choctaw Census Card 4890).
It is here that we find the first problems... Sophronia E. Nail is 9 in 1899 and 12 in 1902. This puts her birth at around 1890, 8 full years after Saphronia and Rufus' first child Martha Alice Stewart in 1882. Similarly, she was living in Pittsburg County, a few counties off from the Hempstead County, Arkansas and Red River County, Texas. Thus I think we can definitively conclude Sophronia E. Nail is not the same person as Saphronia E. Arrendale.
I was also unable to find any other Sophronias or Saphronias that fit the needed requirements to match our Saphronia. So at the least, that can be crossed off the list.
What of the "another little Choctaw" comment made of Margaret Azilee? Hard to say anything concretely, though I believe it was said (it's hard to otherwise explain her Choc nickname), albeit perhaps not with the tone that's been passed on. Perhaps it was said as a comparative as opposed to a declaritive statement on familial lineage? I certainly lean to that given the lack of evidence.
But, what if there was an Indigenous Peoples heritage through other means? This is certainly possible, and would be difficult to both confirm or deny without much more exhaustive research. Still, based on DNA testing, we can come to some general conclusions.
The following family members have been DNA tested to my awareness: Charles "Charlie" Wayne Stewart, Marvin Earl Stewart Junior, Daniel Evan Stewart, Kristen Leigh Stewart, and Michelle Erin Stewart (Saphronia's great-grandson, great-great-grandson, and great-great-great-grandchildren, respectively). In all of these, there is substantial European heritage evident (Irish, English, German, Scandanavian), with some small outlying heritages (Basque, South-western Africa). None of these, however, include any Native American ancestry. Even if the Choctaw heritage came through the earliest known Stewart on the patrilineal side, Charlie Stewart should still be roughly 1/16th Choctaw (erratics of genes nonewithstanding). This doesn't surface, however.
In closing, I have no evidence that supports any substantial amount of Indigenous Peoples ancestry, Choctaw or otherwise, in the Stewart (and by extension, Dollarhide) family tree. If any exists further back on the tree, it is now at non-existent levels. On the topic of ancestry anxieties attached to this, I hope those can at least be partially alleviated with this writing.