If I were a TD at this tournament, here is how I would rule in this situation after speaking with both opponents separately and getting their account of what happened (assuming what both said lines up with what's given in the question), and why I would rule that way:
Telling an opponent during a rated game that they should resign is at best rude, and at worst could be construed as a violation of rule 20G - "It is forbidden to distract or annoy the opponent in any manner whatsoever". I would give the benefit of the doubt to White that they were not intentionally trying to violate that rule.
With that in mind, even if the communication was not completely understood, I would rule that anything that White said to Black, in which both players agree that they said/heard the word "resign", stopped the clock and put away the set, would be reasonably interpreted by Black as a resignation by White, and would adjudicate the game in favor of Black. 0-1.
Disclaimer - I am a certified USCF TD, but only Club level, so I could easily be wrong here. This wasn't on the exam! :-)
[EDIT: After the later follow-on discussion, I understand now that this isn't the correct ruling, and that a better ruling would involve having the players resume the game. I appreciate the patience of those with more experience than me in explaining what they'd do, and why.]