You are wrong. The peaceful part of Islam will be forced to join the war on us or be killed.
The present "small part of Islam that is violent" is ruthless enough to carry the fight to anyone that will not buckle under
Two Points. 1) The small, violent part of Islam is ruthless enough...AND unless we have the stones to kill off every last one of them, they will do what you say. 2) The "majority"..."peaceful" part of Islam is an accomplice to the minority by their silence and lack of standing up to this ruthless generation. Therefor, part of the blame for all of this is on their shoulders for not stopping this before it started.
I personally don't think they are as innocent as they make themselves out to be. I just think they are not as brave as their jihadist brothers...yet. I do believe many are with them in spirit and will join them once the waters are a little less icey.