I don't force apologies, but I do try to help my children see when something needs to be fixed. It's probably easier to explain with examples...
DD says DS just ran her over. I cuddle her, etc. DS comes in seemingly ignoring his crying sister; really, though, he already feels so bad he cannot even look at her to see if she's hurt. (I think this is really common in children and often misconstrued as insensitivity.) I say, "DS, your sister might like a cherry stone pillow from the freezer." He dashes off, comes back as her hero, and suddenly is in the "right" and able to say all sorts of sweet words that moments before I could not have "forced" him to say.
Or, DS comes in saying DD has knocked down all his things. DD says he did such and such first. DS says he only did that because she did such and such. I say, "Wow! you're both pretty upset. What would you like to happen?" Usually, not always, they then work it out themselves. Sometimes, it involves me helping to reset the stage for whatever the game was, but most of the time, they would rather just go back to their play after they've cleared the air.
My littlest does not fit in these paradigms. She is about 3.5, and she is still enough part of me that she would much rather feel my love than a sibling's apology. I act as the mediator when she needs it, but in this role, I very carefully am showing her how to help make things right, or how her siblings are trying to make things right by her.
It sort of sounds like no one ever apologizes, and that's not true. However, I actually prefer actions to the meaningless phrase, "I'm sorry." I DO see how it is a social cue, and I have pointed out to my older two when these words would help to smooth things over. It's just that I'd really rather they acknowledge and fix the problem than think these words are the only thing necessary when we've made a mistake.