Met some women who never ever say sorry so it's not just a male thing to never apologise for anything. My dad has never said sorry to my mum and they have been married for over twenty years, but that particular relationship works. My dad is definitly old school and a real man's man. I don't totally agree with it but it works for him.
If it is important, if you know you done something seriously wrong and you have really upset a woman and your gut, morales, etc leaving you feeling as if you show say something, then by all means say you are sorry and mean it.
But don't say sorry for every little thing!!! There are guys out there that spend their life saying sorry. I been trying to get my mate to stop apologising over every damn little thing. Drives me nuts.
Having a principle of never saying sorry to anyone is all well and good but I think common sense, like most things in life, is way more important. It sometimes takes a lot of courage to put aside your pride and make a humble apology when normally you wouldn't even contemplate doing such a thing - just make sure they genuinely deserve it!
Use your common sense and develope some social intelligence.