McCoy does not argue with Spock because of racist reasons. He does not believe that ignoring your emotions in favour of logic is a good idea, and will clash with Spock. He uses racial slurs as shorthand when he argues.
Agreed, he is not arguing to insult him over his race, but imagine doing that today. You disagree with someone over a topic so you throw in racial comments ? Unacceptable.
Love Bones, it was the times, does not hold up well in 2013.
I recommend you stop looking at it with today's values (ahem), and simply enjoy the show for what it was.
You'll be happier. Trust me.