I thought you were a female as well. Well grats to you and your SO on your new baby girl. Im very happy that the baby is ok after such a panicky beginning. I wish the very best of a recovery, both physically and emotionally to your SO. That being said, while you dont need to sue, definitely getting the word out so that others know, both in and out of the hospital is a good way to go. Not for vengence, but for knowledge. Mistakes can happen, but being carved up without the proper pain meds, and the pain meds being given potentially hurting your child is something that no one should ever go through. There are several nurses and doctors surrounding a woman in labor for a reason; to ensure the best medical care possible and to catch what another person may have missed. I would go to the top in the hospital and make an official complaint. It also does not hurt to see a lawyer and find out your rights on the off chance that long term health issues may pop up.
Research all avenues, but most of all, enjoy your new child, and be a strong support system for your woman. She will most definitely need it.