Tzeitl and Hodl aren't exactly following tradition when they marry. No arranged marriage, no going through their parents to meet their husbands, and no real considerations of money or property before taking the plunge. Tevye has some doubts, sure, but all in all, he's a pretty accommodating dad when it comes to letting his daughters do whatever. But when Chava tries to do the same thing with a guy who is not Jewish, and is instead part of the oppressive majority culture, Tevye cannot handle the betrayal. It's cultural treason.
So here's our answer to the question of just how much tradition it's okay to leave behind. Arranged marriages? Sure, dump 'em. Judaism? Not so much.