Can you apologize without truly accepting any blame? Some CEOs, politicians and celebrities do it all the time. They apologize for the situation, express regret that the wronged party reacted the way he did or wistfully speak about ways the situation might have been different. None of that constitutes an official apology. If circumstances don't permit a face-to-face apology, you can find ways of apologizing without stooping to non-apology "apologies."
Recognize the other person's point of view. Accept the ways in which your words or actions hurt him. Even if you don't speak to the other person about it, know in your own mind that what you did was wrong. It makes your regret genuine and your efforts to make amends more than just empty gestures.
Refrain from excusing your behavior. An excuse is not an apology, and can even make things worse. If you're prepared to apologize—even without saying so—then you're willing to own up to the consequences with sugarcoating the situation.
Accept the other person's anger. This can be difficult. But if you're prepared to listen to the other person tell you about her anger and the ways in which your behavior hurt—again, just listening, without apologizing—it permits them some measure of catharsis. That paves the way for patching things up, and makes it easier for the other person to forgive you.
Make things right. If your words or actions created a bad situation, do what you can to clean it up. Smooth things over with the client, provide compensation for the damaged or stolen goods, or make restitution in some other way. Again, you can do this without having to say "I'm sorry." But the effort does count, as long as it is genuine.
Don't repeat the mistake. Whatever you did or said, don't do it again. If you avoid your earlier actions, you will show that you're aware of the severity of the situation and realize that it hurt the other person. Going out of your way to be nice to the person may appear patronizing, but after things have quieted down, you may wish to make some gesture to restore her good feelings. Buy her lunch, let her take off from work early or pick up a little gift for her on a day when she's feeling down. It makes a difference.
While you can legitimately apologize without saying "I'm sorry," beware of the non-apology apology. Expressing regret about "the situation" or saying things such as "I'm sorry you feel that way" is disingenuous and often makes the situation worse. An apology assumes responsibility for the situation and endeavors to make amends, regardless of what form it takes. Anything else is just another kind of evasion.