P.M.
It may not make sense to her (it doesn't to me) to require a child to say she's sorry. If she's not sorry, then we're teaching her to lie. If she is sorry, she will need to see apologies being practiced by the adults in her life in order to learn the beautiful art of the heartfelt apology.
The crying could actually mean quite a few different things, so you might have to consider the bigger picture to figure it out. It could well be that your daughter feels contrite. Or it could be true that she's simply embarrassed or angry that she got caught. If it happens a lot, she could feel that she lets you down all the time, and feels insecure about your love.
My grandson is 4, and with a little gentle coaxing, is able to tell us quite a bit about what he's feeling when he gets upset. We do have to give him the opportunity, though.
Good luck. I'm glad you're asking. I hope you find a gentle solution and don't let this escalate into a more serious problem.