Despite my very gloomy view of the value of international input into the Middle East conflict when it turns hot, I think that the EU has the ability to play an important part in the peace process. In the end, change must come from the Israelis and Palestinians themselves, but the EU can and should play a role in creating conditions which would encourage this.
There is a very interesting article in EurActiv on this topic.
The EU finds it very hard to coordinate foreign policy in this area, as can be seen from the argument over whether or not to call for an investigation into possible war crimes committed in Gaza and into the bombing of UN buildings by Israel.
As formulating common policy from crisis-to-crisis is almost impossible, the EU should play to its strengths. The EU's power and influence is mostly economic and institutional: the size of the single market gives it economic power concerning access and entry, and institutional make up of the EU gives it the attractive political lure of membership, etc.
So a more successful, and possibly more easily sustainable, EU policy on the peace process may be to make a massive commitment to the region based on the common end goal of the two-state solution. An "objective" institutionalised mechanism for engagement would offer a more coherent and effective policy towards the region. It would be objective not ideologically - it would be clearly committed to the two-state solution - but "objective" in that it would involve a progressive ladder of incentives and sanctions to which the member states would be committed, and which would reduce the scope and effects of differing political viewpoints based on recent flare-ups in the conflict.
So the EU could commit to a system of rewards and punishments for progress on peace, based on negotiations leading to a two-state solution and on economic links and security between the two states. For Israel, incentives would include better relations with the EU and more access to the single market, and sanctions would be the lessening of such links and access. For Palestinians, development aid and state building would form the basis of the incentive/sanction scale. If eventual membership is a possibility, it needs to be placed in the context of a successful peace that is proven to be sustainable.
The Quartet could also participate in a more institutionalised peace process, but the EU should not leave it completely to others to determine its approach to the conflict.