A case can be made for the Prince having to assume a good amount of responsibility for the tragic turn of events in the drama. Essentially, Shakespeare shows the political leadership of the Prince as limited by the factionalized notion of power that the families possess. The Montagues and the Capulets hold economic and social power over Verona. The Prince is merely a political prop in the face of their feud. The Prince is well aware of the blood rivalry between both families. The opening scene is one in which violence between both families fills the streets. Yet, the Prince does not do more than offer a stern warning to both Capulet and Montague. The Prince fails to construct a lasting peace when Romeo has murdered. Rather, he banishes Romeo.
The Prince ends up concluding at the end of the drama that a lesson has been learned in which "all are punished." Yet, I think that some level of blame can be assigned to the Prince as not having anticipated that such a lesson can be gained without the sense of sadness that preceded it. Shakespeare constructs political leadership as one that must envision the right path even if it comes at the cost of those in the position of economic and social power. For all practical purposes, the Prince does not challenge those in the position of power. In not challenging these forces, some level of blame has to be assigned to the Prince for what happened.