Many people say that having your children say "I'm sorry" when they aren't really sorry is invalidating. I am certainly against invalidating your kids, and I do not recommend telling your kids that they ARE sorry when they aren't. However, I do think that children should be encouraged to apologize if they hurt someone physically or emotionally, and here is why.
I have noticed how often I -- and my female friends -- apologize for things that are out of our control. This, of course, is different than saying, "I am sorry for being late," or "I'm sorry I picked a fight last night because I was really hungry and you were taking forever to figure out our plans."