It was meant to be over by now. This time last week Israeli military planners were demanding another 72 hours to finish the job: that's all they needed, they promised, to clear southern Lebanon of Hizbullah. Yet the enemy has proved stubborn. Despite two weeks of bombardment, Hizbullah's formidable arsenal remains in place. Yesterday they fired yet more rockets - 60 of them - deep into Israel, reaching the city of Haifa and killing a teenage girl in the Arab village of Maghar.

This persistence is causing the first rumblings of Israeli disquiet. Why are the Katyushas "still coming, and killing?" asks one Israeli columnist. Are the Israel Defence Forces losing their edge, asks another, wondering if "instead of an army that is small but smart, we are catching glimpses of an army that is big, rich and dumb". The top brass deny they have been surprised by Hizbullah's strength. They expected nothing less, they say - not least because Iran has been supplying the movement with more than $100m worth of arms. Which would explain the serious hardware, including long-range missiles, at Hizbullah's disposal.

So far none of this has eroded the astonishingly high level of Israeli public support for the war. I spoke yesterday to a "refusenik", an Israeli soldier whose principles compelled him to spend a month in jail rather than serve in the West Bank or Gaza. Even he was clear: "We had no choice but to hit back." This is not about defending occupied territory, because Israel is not an genuine occupier in Lebanon. This is, he says, about defending the country from a proxy army of a state, Iran, that is committed to Israel's destruction.

Little has punctured Israelis' sense of self-belief. They see few of the TV pictures we see, showing Lebanese children, bloodstained and weeping; they have victims of their own to concentrate on. As for the rest of the world's condemnation, it doesn't cut much ice. Why should Israelis listen to Vladimir Putin when he tells them their response has been "disproportionate"? Was Russia's pounding of Grozny proportionate? As for complaints from Britain and Europe about the 390 civilians killed in Lebanon, those are a reminder of the more than 3,000 civilians killed in the 2001 onslaught against Afghanistan: how was that proportionate exactly? Kim Howells was right to be appalled by what he saw in Beirut. But he surely would have been just as shocked had he visited the Iraqi city of Falluja after the Americans had turned it to rubble.

Besides, not much of this criticism, including that from Howells, has got through at all. The message projected by most of the Israeli media is that the bit of the world that matters - the US - is behind them. The government certainly echoes that line, and it will have been emboldened by Condoleezza Rice's show of understanding yesterday.

Indeed, for prime minister Ehud Olmert the backing of the US is central to everything this war is about. The Tel Aviv University analyst Dr Gary Sussman calls it a "war for the legitimacy of unilateralism". This approach, first pursued by Ariel Sharon and now Olmert's defining project, tells Israelis that it is OK to pull out from occupied territory - whether southern Lebanon in 2000 or Gaza in 2005 - because after withdrawal there will be a clear, recognised border, behind which Israel can defend itself more vigorously than ever. That is why, once Hizbullah had captured those two Israeli soldiers, Olmert had to hit back. If he had not, he would have vindicated the critics who brand unilateral withdrawal a glorified retreat, jeopardising Israel's security. He had to prove that pulling out did not mean running away, that Israel could still defend itself. What's more, because it had moved back to the internationally recognised border, Israel would now enjoy international legitimacy. Washington has obligingly played its role, supplying the support that confirms Olmert's logic.

This message is not aimed solely at the Israeli people. It is also meant to restore the country's "deterrence", telling Hizbullah and the rest of the region that they cannot cross Israel's borders, or seize its personnel, with impunity (no matter how Israel itself behaves). Israel is especially keen to disprove the "cobweb theory", put about by Hizbullah: pull at one Israeli thread, such as its 18-year presence in Lebanon until 2000, and the rest will unravel. The current operation is designed to say that Israel does not do unravelling.

There is a last audience for this war. Olmert wants the Palestinians to see that if Israel withdraws from further territory, as he intends, it will not be a soft touch. On the contrary, as the world has seen, if Israel is so much as scratched it will hit back very hard. The prime minister wants this point seared into the minds of Hamas and Fatah so that they remember it come the day Israel withdraws from parts of the West Bank.

From his own point of view, Olmert had little alternative. If he had accepted the soldiers' kidnapping, and sought their return through diplomacy, most Israeli analysts are agreed that he would have been finished. He would have confirmed his own weakness, a civilian with no military record, and he would have proved the anti-unilateralists right. His own plan, to withdraw from more occupied territory, would be in shreds. As things stand, he should now have the credibility to move forward.

That's as close as we get to a crumb of comfort to be found in the rubble of this last fortnight. Yet it need not have been this way. Had one of the key players in the drama behaved differently, this entire mess could have been avoided.

I'm thinking of the United States. It's fashionable to blame the US for all the world's ills, but in this case the sins, both of omission and commission, of the Bush administration genuinely belong at the heart of the trouble.

Diplomacy has had a difficult task from the start, in part because the US is not seen as an honest broker, but as too closely aligned with Israel. Washington has long been pro-Israel, but under President Clinton and the first President Bush there was an effort to be seen as a plausible mediator. Not under George W. Far from keeping lines of communication open with Hizbullah's two key patrons - Syria and Iran - they have been cast into outer darkness, branded as spokes, or satellites, of the axis of evil. As a result there has been no mechanism to restrain Hizbullah. Now, when the US needs Syria's help, it may be too late. Damascus will extract a high price, no doubt demanding the right to re-enter, in some form, Lebanon. The White House can't grant that - not when it considers Syria's ejection from Lebanon in 2005 one of its few foreign-policy successes.

But the record of failure goes deeper than that. It began in the president's first week, when Bush decided he would not repeat what he perceived as his predecessor's mistake by allowing his presidency to be mired in the fruitless search for Israeli-Palestinian peace. Even though Clinton had got tantalisingly close, Bush decided to drop it. While Henry Kissinger once racked up 24,230 miles in just 34 days of shuttle diplomacy, Bush's envoys have been sparing in their visits to the region.

The result is that the core conflict has been allowed to fester. Had it been solved, or even if there had been a serious effort to solve it, the current crisis would have been unimaginable. Instead, Bush's animating idea has been that the peoples of the Middle East can be bombed into democracy and terrorised into moderation. It has proved one of the great lethal mistakes of his abominable presidency - and the peoples of Israel and Lebanon are paying the price.

· freedland@theguardian.com