The purpose of a apology is to express genuine feeling of regret and to take responsibility for the hurt you caused another person. To simply require a child to do that by rote ("say you're sorry) takes away from the bigger goal of helping them learn to identify and process their own feelings and learn to accept responsibility. Forcing "I'm sorry" is nothing more than teaching kids to impersonate genuine feelings of remorse and that faking can become a habit of lying that carries over into other situations. Tell then what they want to hear so they shut up.
Of course your daughter isn't a sociopath and of course she doesn't want to disappoint you. That's all the more reason to allow her processing time and to come to the decision to apologize rather than forcing it on her.
For all kids I would hope that their primary motivation for good behavior becomes about something other than disappointing mom. If not, that's a pretty weak foundation to enter the teenage years when kids may seek to differentiate by doing the opposite of what parents want.
