Actually, the Empire started the civil war, by deliberately interfering in Nord custom after their puppet lost the challenge. Had they not done so, the Moot would have proceeded and a new High King (likely Ulfric, due to how the rules of succession work) would have been chosen, in accordance with ancient tradition, which would have removed almost all of Ulfric's justifications for turning on the Empire in the first place. Without that he would have been unable to rally support for breaking away, which would have averted the civil war entirely.
Had Ulfric been chosen (as was likely) he would have moved to secede from the Empire anyway, although it would have been a significantly harder sell and might actually have failed. Instead, it probably no longer matters who gets picked; the Empire's attitude toward their provinces has been exposed for what it really is, which pretty much guarantees a permanent break.
A moot is convened every time a High King/Queen dies to determine which member of the royal family is suitable for the throne, and that's also how Torygg was chosen. Ulfric was at the moot to select a new High King/Queen when Torygg was selected, and even then he raised hell over it. When he didn't get his way, he decided to take action and killed Torygg since he got frustrated using just words. I can't imagine that, if they did choose to go to a Moot, it would've went well if the Empire didn't intervene. Ulfric was liable to just kill everyone who didn't agree with him if he took out Torygg, and Skyrim was still under the Empire's protection. The only way Ulfric would've become High King after killing Torygg would be because he threatened or scared the other Jarls. So Ulfric kinda is the source for all the unrest, no matter what his reasoning is. If he just played dead or chosen to bide his time or not done things to the nth degree, there'd be no Markarth incident, no dead High King, no civil war.