How do I reply to "I'm Sorry?"

I can only answer on the basis of the spoken word so cannot give references as is usually done on this site.

The pseudo-question “I’m sorry?” is used in the sense of “I heard what you said but do you really mean it?”, “What? Do you expect me to believe that?”, “You’ve interrupted me. What do you want?”, or “I don’t agree with you. Explain yourself”.

It is often often delivered slightly disdainfully, as if the speaker can hardly believe there is any possible explanation for the statement or interruption.

The best reply is to act neutrally and politely and to answer on the basis of one of the above understandings.

Edit: I am grateful for comments that add the idea that Excuse me? may be simply a request to repeat what was not heard.

It's a question as if to ask if they will accept the apology. Using the form of telling a child to do or state "I apologize" is the action of or the verb form as you are committing the act of apologizing. Your original phrase "I'm sorry?" is a statement of offering a possession or feeling. Kinda like giving your gratitude, would make it a noun as it's not physical or material, it's still a thing. Just like anything else, you can't force someone to accept or receive something. At least, it isn't good manners. Especially, if you've already wronged them in someway for which is the ultimate reason of offering your apology in hopes that they'll accept it by following it up with "forgiving" that individual most commonly with one of two phrases. Either "I accept" or I've always preferred, "Your forgiven".