Essentially, Dr. Miller and Josh Green represent the two poles of a debate among African-Americans in the wake of the Wilmington "race riots," the forcible overthrow of African-American leadership in the city of Wilmington, North Carolina, in 1898. The events in Marrow of Tradition are based on these events.
Dr. Miller represents a moderate voice among African-Americans. He is educated, is married to a woman of mixed race, and belongs to an emerging black middle class. He believes open resistance to white supremacy is futile, a conviction he displays when he is forced to switch passenger cars on a train in Virginia, and most tellingly when he refuses to lead the black community in armed resistance against a lynch mob.
Josh Green, on the other hand, is angry and bitter, having lost his father to Klan violence (specifically at the hands of Captain George McBane) when he was young. He believes in violent resistance, even if it results in his own death. Having much less to lose than Dr. Miller, he is far less patient than the doctor, who is a believer in gradual, orderly reform. The differences between the two are highlighted in an exchange in Dr. Miller's office, when Greene shows up for treatment after being wounded in a fight. "You'll hit the wrong man one day...You'll get into a quarrel with a white man, and at the end of it, there'll be a lynching, or a funeral."
Greene replies, referring specifically to McBane:
I expec's ter die a vi'lent death in a quarrel wid a w'ite man...and fu'thermo', he's gwine ter die at the same time, er a little befo'...I'm gwine ter kill dat man as sho' as I'm settin' in dis cheer...
His prediction comes true, but Miller's nobility in rescuing Carteret's son near the end of the novel confirms that while Chesnutt admires the courage of Green, he sees true nobility in taking the high ground.