Learn Dutch: Ik vergis me, sorry. - I am making a mistake, sorry.

The verb zich vergissen (to make a mistake / to be wrong) is a reflexive verb. This means the action reflects back on the subject, so it always requires a reflexive pronoun like me (myself), je (yourself), or zich (himself/herself). In Dutch, you literally say "I mistake myself."

Could I use the stressed pronoun mij instead of me?

No, it sounds very unnatural here. We use stressed pronouns like mij when we want to contrast or emphasize who is receiving the action. Because you can only ever "vergis" yourself, there's no reason to emphasize it, so we always use the unstressed me.

The English translation says "I am making a mistake." Can I just say Ik maak een fout?

Yes, ik maak een fout is grammatically correct and means the same thing. However, ik vergis me is the most common and natural way Dutch speakers say "I am mistaken," "I've got it wrong," or "My bad."

How would I change this to say "He is making a mistake"?

You would say Hij vergist zich. The verb takes the regular -t ending for "he," and the reflexive pronoun changes to zich (himself).