IMO, if you spank you fail as a parent (not fail forever, just fail in that instance). And I spank. Spanking should be a last resort. A tool to be used when all others have failed. Physical pain should only be employeed when the child needs to associate physical pain with a certain action, and they cannot understand the reasons or the repercussions. For instance, I might tell my 2 year old to not play in the street. She does not understand what getting hit bya car feels like, or what it may mean for the remainder of her life. I can also put up a fence. I can make sure I keep a close eye on her when she is outside. But when all my precautions fail, and she plays in the road, I spank her. The physical pain of a spanking is to avoid the worse pain and repercussions of a car accident. My failure as a parent of spanking is preferrable to the bigger failure of having my child get killed or maimed by a car.
In your story there was nodescription of other steps you took to talk to your child. Maybe you just left them out, but just spanking and restrictions do not address the issues. Why did he call her a bitch? How did he want her to feel? How does he think she does feel? Where did he hear that language? What makes him think it is okay to use this language in your home at all?
Whenever you have a problem, you need to have 2 solutions. One for the immediate effects, and another for the root cause in order to reduce the likely-hood of the problem occuring again. No offense, but it dosen't sound like you have addressed either.
Sticking by your guns is good. But so is admitting when your were wrong. It dosen't matter which you do, as long as you feel good about it - and as long as you do both whenever they are needed. If you do back down, tell him exactly WHY you are backing down so there is no perception that he can manipulate his way out. You punishment does seem a little harsh if this is a first time offense and if there have been no previous discussions about acceptabel language and name calling. It would be a good exercise to have your family sit down together to develop your own family rules and for the whole family to agree on the appropriate disciplines.
Goodluck!