Yes and no.
Many people use the expression "I'm sorry" to express their regret about something they've done and to begin the process of making amends to the person they've hurt. But when we look at the expression closely, there is a subtle difference between saying "I'm sorry" and "I apologize."

The word "sorry" is an adjective that describes how you feel. That is, you feel sad that you hurt someone. But does that mean that you are asking for forgiveness and willing to make a change? That's where the verb "apologize" can be more explicit in helping you to show your desire to make things right again.

To apologize means that you are doing more than just feeling something, you are acknowledging the impact that your mistake had on another person. It is about you taking an action (apologizing, giving an apology) directed outwards to the other person.
In short, saying you're sorry merely expresses a fact. It is about you and your internal state.

So if you want to make sure there's no confusion between how you feel and your intentions to make the other person feel better, try saying: "I'm so sorry I hurt your feelings. I apologize for being so thoughtless and inconsiderate and I will not [describe action that you are apologizing for] again."