I have read a short story for the 2nd time and still have a problem with the last few sentences:

Cosí la lie che il Re voleva evitare scoppiò piú aspra di prima, e il Re per farla finita, decise di sposare la figlia a un quarto pretendente, che era venuto a mani vuote.

I think it may mean that the king decided that each potential suitor can half 1/4 of his daughter??? I'm probably totally off!

Thia is from un fiabe Italiane, "L'uva salamanna."

It means that the King decided that the one that should marry his daughter was a fourth suitor that did not bring anything to him/them...
Does this make any sense considering the rest of the story?

Yes, it makes sense. What does "pretendente mean?" And I can't remember what "mani vuote" means? Thanks!

A mani vuote literally means "with empty hands" i.e. empty-handed.
Pretendente is a suitor or admirer.
You can find a good English-Italian dictionary here