Opinion: After Gaza, will Barack Obama push for peace?

All sides involved in this month's Gaza crisis face an array of choices. They might simply claim victories, as Israel and the Palestinian Hamas faction are doing. They could feel the satisfaction of having avoided a wider war by mediating the cease-fire -- a credit claimed by the United States and Egypt.

There is the likelihood, however, that while the Israelis and Palestinians repair the damage they suffered in their eight-day conflict, they will mostly be preparing for the next round of bloodshed. But the Obama administration is in a unique position to reorient the trends in the Middle East, so that the Gaza events lead to a new push for peace.

President Barack Obama demonstrated that the United States is the irreplaceable go-between when he dispatched Secretary of State Hillary Clinton to Jerusalem, Ramallah and Cairo. She was able to preside over the announcement of a cease-fire on Wednesday in the Egyptian capital, where she spoke of a need "to consolidate the progress" and improve the lives of Israelis and Palestinians.

Strengthened by his re-election, Obama may choose to renew the American mediation effort that led to disappointment in 2009 and 2010. This could involve not only prodding Israeli and moderate Palestinian leaders to sit around a negotiating table, but a wider goal of creating a pro-peace, pro-stability coalition in the Middle East.

At their most ambitious, officials and diplomats speak of fashioning an axis of Sunni Muslim nations that want to diminish Shia influence and to stand against Iran's nuclear ambitions and support of terrorism. The axis might naturally include Egypt, Turkey, Saudi Arabia and Qatar.

More specifically, as an outgrowth from the Gaza crisis, American officials focus on three key leaders: Egypt's President Mohammed Morsi, Turkey's Prime Minister Recep Tayip Erdogan and Israel's Prime Minister Benjamin Netanyahu.

The officials say Morsi came through last week. As a Muslim Brotherhood member, he was expected to sympathize entirely with the radicals of Hamas. Yet, obviously aware that Egypt's economy is in terrible shape -- with an almost total absence of tourism revenue -- he saw the value of billions of dollars of U.S. aid. Morsi helped the Obama administration by toning down the passions and finding a practical path to a truce.

This being the Middle East, there can always be unexpected actions and reactions, as we now see in Egypt. Morsi decided to take advantage of the prestige that Gaza diplomacy gave him and issued decrees on Friday that give him powers superseding ordinary laws and the judiciary. As thousands of protesters again filled Tahrir Square in Cairo, the Obama administration expressed "concerns." But it seems to be tolerating Morsi's moves.

As for Turkey, Erdogan has been saying hostile things about Israel for years, but Western diplomats in the Middle East continue to hope that a strategic partnership, which used to be a huge plus for Turkey and Israel, can be restored. The diplomats note, with satisfaction, that the heads of Turkish and Israeli intelligence were both in Cairo while the Israel-Hamas cease-fire was being negotiated. Considering that Turkey and Israel share deep concerns about Iran as a rival power center -- and about the possible takeover of Syria by anti-Western extremists -- they may well find a lot of ways to cooperate again, both covertly and openly.

The U.S. government could help its own pro-stability push by mediating an end to the bitter argument over the killing by Israeli troops of nine people on a Turkish aid ship that was heading for Gaza in 2010.

The hope voiced by people -- dwindling both in Israel and in Muslim countries -- who have somehow retained their optimism about peace prospects is that a pro-stability Washington-Cairo-Ankara axis can support positive developments and factions in the Middle East, while working together to oppose radicalism.

That could give a much healthier background to efforts to renew Israeli-Palestinian peace negotiations. And success would strengthen an anti-Iran coalition.

The talks would face many obstacles. Hamas consistently refuses to speak directly with Israel, and vice versa, so the best to be expected would be a deal with the moderate Fatah faction in the West Bank.

Yet there are many strong reasons for the Obama administration to try again. Among its arguments to a skeptical Netanyahu? If you want U.S. backing for whatever may unfold in the Iranian nuclear crisis, cooperate with the pro-stability axis we are creating. And, while you are presenting yourself to voters on Jan. 22 as a tough, security-minded leader, please consider that nothing would make Israelis more secure than an overall settlement with their nearest Arab neighbors.

Dan Raviv, a CBS News correspondent in Washington, and Yossi Melman, an Israeli journalist specializing in strategy and intelligence, are the authors of "Spies Against Armageddon: Inside Israel's Secret Wars." They blog at IsraelSpy.com.