The phrase 'Ich entschuldige mich' is an abomination. The literal translation would be 'I excuse myself'.
Instead, you're supposed to say 'Ich bitte um Entschuldigung' (I beg your forgiveness) - thus you're asking the person who was wronged for forgiveness. It should not be the choice of the wrongdoer to decide whether to be forgiven or not. Though to be fair, commonly it's used just as you described.
If I remember correctly, Kraus has at least once written about this specific phrase.