Does the Hadith "A Muslim should not abandon his brother for more than three [days]" apply to someone who cut ties with me because he thought I wronged him, even though there was no prior relationship?

It is obligatory upon you not to intentionally abandon her and refrain from greeting her with "Salam." Most scholars agree that abandonment is removed by giving "Salam." So, if you greet her with "Salam" when you meet, you have fulfilled your obligation. If she refrains from returning the "Salam," the sin is upon her, not upon you.