Depending on the severity of the transgression, a "sorry" - no matter how purely intentioned - isn't going to be an instant salve.
The only thing that I've found to be a surefire fence-mender is time. If you truly feel bad, you'll serve your time in the doghouse, mind your P's and Q's and, eventually, things will be forgotten by all but the most vindictive of assholes.
Don't apologize with your words so much as your actions. Do everything possible, do it right, and walk the straight-and-narrow without calling attention to the fact that you are doing so.