I used to work for an Israeli company, I loosely keep in touch with several folks there. The "average" Israeli, according to most of them, wants peace and would gladly accept a two state solution.
The key as to why many still hesitate, and don't want to just make a formal state of Palestine, is that the Palestinians have proven NOT to be able to push out the violent part of their population.
There is a history in Jordan, Kuwait, Egypt, and I think even Saudi Arabia, where Palestinian populations - or that extreme element at least - incited violence, caused strife, and celebrated local Arab deaths. There is a reason that Palestinian refugees are not being taken in by other Arab countries....
These Israeli's I know agree that Israel will gladly give land for peace (TBD about Netanyahu), but still would need proof that Palestinians would be peaceful before allowing them to be a formal state. Self-govern is indeed one thing, and in the case of Gaza (which has self-governed since about 2005) it has not proven peaceful at all.
One person I know was in tears over the bombings, for the civilian lives lost, and as Iain says will be propaganda for Hamas recruitment. But he also echoes others' thoughts that Israel almost has no choice. If they give in too much without the promise of peace, they will be seen as weak, and others will repeat what Hamas has done.
I just saw a picture by Banksy which showed a small Jewish and (I assume) Palestinian child watering/caring for a small tree that seemed just planted. That made me sad...if one generation can be taught that peace is an option, there can be hope. But so far, I'm not seeing it....