It's hard to disprove a historical might-have-been, but there were Indians fighting on the Confederate side in the battle of Pea Ridge (March 1862). So there was a little Indian killing of Unionists in northwest Arkansas in the war.
There were also Indians fighting on the Union side. In fact, Grant’s aide who wrote out the final terms at Appomatox was a full blood Seneca.
The Cherokees fought their own amazingly bloody little civil war in the middle of the bigger war.