Apologising doesn't have to be an admittance of guilt or regret. If you've done something that hurt someone you care about, you can apologise just to let them know you care.
There's a difference between apologising for your actions and apologising for how those actions hurt someone else. Sometimes a simple "I'm sorry" can go a long way. This is a small price to pay to repair a friendship or relationship, so next time you're struggling with your sense of pride, drop it and give a good apology.
Apologizing Means That You Value Your Relationship [Wisdom Quotes and Stories via Dumb Little Man]