A real wildcard here that we don't fully understand is how powerful Ulfric really is.
When I played through the game the first time, I got a sense that one of the main reasons the Imperials hadn't launched a full-scale invasion of Stormcloak territories (Windhelm, in particular) to suppress the rebellion was that they were wary of engaging the Stormcloaks in wide-scale, open-field combat. I agree with what other posters have said regarding the Imperials' military advantage (sheer manpower + training + weapons + tactics + other factors), but if Ulfric's Voice (which we really know VERY little about, other than that he has at least one shout and he used it to kill Torygg) is anywhere near as powerful as the Dragonborn's... he could be DEVASTATING in a large-scale battle. A kind of superweapon, potentially, able to thwart any Imperial attempt to fight the Stormcloaks head-on. We can assume that he is less powerful than the Dragonborn, though, but if Ulfric can turn the tide of any battle with his Voice it would explain both the Imperials' hesitance to crush the rebellion AND why the Stormcloaks don't seem to be accomplishing much with him sitting in his palace and yet are still portrayed as some kind of rough equal to the Imperial contingent present in Skyrim.
All of that leads me to believe that if Ulfric dies, the Stormcloak cause is pretty much done for. The movement already appears to be very centralized around Ulfric and his cult of personality... it's not like the real-life example of Washington in the American Revolution, who would have simply been replaced if he had died, albeit by someone of lesser tactical ability and charisma. Most of the Stormcloak rhetoric centers on Ulfric and making him the High King.
Most of the conversation in-game, from what I recall, regarding the Ulfric-Torygg incident regards the "legitimacy" of Ulfric's claim to be High King of Skyrim. Ulfric himself seems to emphasize that he killed Torygg to prove that HE should sit on the throne... If Ulfric died, then any point the Stormcloaks might have is moot.
The Stormcloak cause, as a whole, seems rather vulnerable in that it is wholly reliant on Ulfric being alive so that he can be the "true High King of Skyrim." There doesn't really seem to be a greater explanation for the uprising, in fact, it is said by a few NPCs in-game that a proto-Stormcloak movement had existed since the ending of the Great War, but it wasn't until Ulfric got involved that it became a full-fledged rebellion and took the form that we see in-game.
Regarding the issue of the Stormcloak insurgency as a "rebellion," "revolution," or "civil war," I think it is, in a way, all three. Stormcloak rhetoric, to the best of my knowledge, asserts that they want Ulfric to be king (the "revolution" part). That in and of itself is not wholly incompatible with Skyrim being a part of the Empire, but it does seem to be implied that an Ulfric-led Skyrim would NOT consider itself a part of the Empire. Skyrim is part of the bedrock of the Empire, via the whole Tiber Septim thing and the fact that Skyrim, along with Cyrodil, are pretty much the founding provinces of it. In Morrowind we saw for ourselves a province that was OCCUPIED by the Empire - how the natives felt, how Imperial citizens/legions were distributed, the history of how Morrowind was forced into the Empire, and how the Imperials themselves viewed the province. Skyrim, both the province and the game itself (atmosphere, lore), feels quite different. The Empire is not a foreign force in Skyrim, we hear this over and over from NPCs and know it from the ES lore.
The American Civil War shares some parallels to the Skyrim Civil War, but is different in several key ways. First of all, the southern states of the US tried to SECEDE from their parent nation to form their own - the war only happened because the northern states viewed secession as unacceptable and acted to forcefully reintegrate them. From what we can reasonably assume, the population of Skyrim is split roughly equally in support of both sides along a few geographical and political boundaries - Solitude, Markarth, Whiterun* in support of the Empire, Windhelm in support of the Stormcloaks, etc.
The Stormcloaks want to FORCE Markarth, Solitude apart from the Empire, even if those cities/Holds do not want to leave. That is what makes them different from the Confederate states, who were perfectly fine with the separate existence of a United States, so long as it let them be their own independent nation. And it's not like the Stormcloaks were a common, unified race, either - we see that the native Nords are split. It's a revolution because the Stormcloaks oppose the current governing body, but it's also a civil war because the governing body, even if it called "the Empire" and has its seat in a different province, is a very natural part of Skyrim.