Yes, he is a crusty old Southern fossil.
The official Star Trek database (at StarTrek.com) uses your choice of words, almost verbatim, in describing McCoy's disposition:
His temperament was sometimes argumentative, a cynic's outer crustiness masking deep caring beneath the surface. His "old South" roots led to the old-time physician manner of doctoring, with a Southern accent that was most apparent when under stress....McCoy played his role as psychologist expertly to the hilt — especially for the ship's two senior officers. As such an emotional watchdog he was not afraid to take on his captain, but it was his running battle of wits with Spock which became legendary. Spock showed his true feelings, though, as when inviting McCoy down to Vulcan for his "wedding" and in storing his katra with him before a known suicidal saving of their ship before the Genesis detonation.
In other words, McCoy and Spock have an underlying respect for one another, but McCoy displays a kind of biting wit and "crustiness" that mask his respect for Spock.
All in all, while humanity has evolved by McCoy's time to have eliminated most public displays of prejudice, this does not mean that every single individual adheres to this, and certainly not in equal measures.
Even by the 24th Century, humanity still has an underlying strain of prejudice. From "Encounter at Farpoint":
RIKER: But your files, they say you're a —
DATA: Machine? Correct, sir. Does that trouble you?
RIKER: To be honest, yes, a little.
DATA: Understood, sir. Prejudice is very human.