I think we should take a moment and understand what an apology looks like when it constitutes true sorrow over an act. I have been seeing a lot of apologies fly around that make a mockery of repentance.
An apology that is truly meant is an apology that says, "I offended you when I(________). I am so sorry that I came across/acted out so badly." An apology that smells of truth owns the offense and humbly admits it to the person offended.
Unfortunately, too many apologies I have heard in the last 5 years go along the lines of, "I'm sorry you took what I said/did the wrong way. You misunderstood me." When your apology makes you the victim, you have not apologized; you have made the offended person the person who is wrong for being offended.
Matthew 18:7 says, "Woe unto the world because of offenses! for it must needs be that offenses come; but woe to that man by whom the offense cometh!"
People are going to be offended, but Christ did not pronounce woe upon the offended; He proclaimed woe unto the offender. If you know you have offended someone, make it right. Just admit the person was offended and be sorry--truly sorry-- that you offended them. It doesn't matter how silly it was. Maybe the offended person was having a bad day; maybe you said something without knowing some of their history and you touched a sore spot. It isn't that person's fault for being offended.
We need to get over ourselves and take care of the offended instead of blaming them for being offended.