Why is "I won't forgive you" a commonly used threat? In a number of shows, "I won't forgive you" or some variation is used as a threat. Rukia uses it when she is trying to keep Ichigo to follow her into the Soul Society, saying, "If you follow me, I will never forgive you." It is used similarly in other shows. In Vampire Knight, Kaien Cross says to Kaname that, if he makes Yuuki cry, he will never forgive him.

This doesn't seem to be used in Western television, or at least not to the same amount as it is in anime. Is this a part of Japanese culture or is this something that originated in anime/manga? If it did originate in anime/manga, where did it first appear?
It's a culture thing, and it's kinda like a fixed phrase, which is not translated accurately into English. If you can understand a little Chinese, its true meaning is 我饶不了你, in which 我(means I) 饶不了(means won't forgive) 你(means you), and the meaning is perfectly translated.

Its true meaning in English is more like this: I will not absolve you from guilt or I will not remit your punishment! (but these are too strong)

For your better understanding, this phrase suggests that if you do anything unpleasant to me, I will remember that one and find a way to make you pay.

PS: Even so it's supposed to be a threat, but this phrase is used between "frienemies" in many cases. For a instance, A doesn't want his partner-like rival B risks his(B's) life to save him(A), then A may say I will never forgive you if you risk your own life to save mine.