Things might get tense in the green room next week at the Walter E Washington Convention Centre in Washington, DC. Among the guest speakers at the annual conference of the American Israel Public Affairs Committee, or Aipac, will be both Hillary Clinton and Binyamin Netanyahu. Normally the US secretary of state and Israel's prime minister could be expected to engage in a mutual love-in. But these are not normal times.

On Friday night Clinton surprised Netanyahu at home with a phone call that amounted to a 43-minute bollocking. Two days later President Obama's senior adviser, David Axelrod, accused Israel of an "affront" and an "insult" to the United States. No wonder Israel's ambassador to the US – who himself had been summoned to the state department for a dressing down – reported that relations between the two countries had slumped to a 35-year low.

What's going on here between two nations normally seen as inseparable, allies so close their enemies depict them as a two-headed beast? The immediate cause of the fallout is the slap in the face Israel gave last week to the US vice-president, Joe Biden. A lifelong friend who is proud to call himself a Zionist, Biden was in the country on a mission to make nice, giving Israel a warm embrace on the eve of a new round of indirect Israeli-Palestinian talks. Biden's welcome gift from the Israelis was an announcement of the planned construction of 1,600 housing units in the Ramat Shlomo neighbourhood of East Jerusalem – even though the US had sought a freeze on all such building in areas conquered by Israel in 1967.

That this was a poke in America's eye, few disputed. Netanyahu insisted that stupidity, not malice, was to blame: a low-level planning committee had made the decision on technical grounds, no offence intended. Biden responded by turning up 90 minutes late for dinner with Netanyahu and his wife, and delivering a ferocious statement of condemnation – but that seemed to be that. Until Hillary picked up the phone.

Why would the Obama team have chosen to escalate a row they could easily have let fade away? "They weren't exactly looking for a fight," says Daniel Levy, Middle East analyst of Washington's New America Foundation, whose ear is close to the administration ground. He notes that Obama is on the brink of passing healthcare reform – and hardly needs to distract attention from that most perilous of battles. The danger will be more acute if pro-Israel Democrats make a "linkage" between the two issues, demanding that Obama lower the pressure on Israel in return for their votes on the health bill.

One explanation is the face-value one: that Obama was "incandescent with rage" at the one-fingered salute that greeted his deputy, and even more furious at Netanyahu's subsequent attempts at an apology. These insisted that Israel had every right to settle in East Jerusalem – but conceded that it was wrong to announce the fact while Biden was in town. This emphasis on timing was, says Levy, tantamount to saying: "I'm sorry I slapped you on Monday: next time, I promise, it won't be on a Monday."

There are other explanations for the US decision to hit back hard. One is that Obama is seizing on the Biden row to send a message to the Arab world: to show that he won't be pushed around by Israel. This view has been given extra traction by a Foreign Policy article reporting that a team of senior officers from US Central Command recently briefed the top brass at the Pentagon, declaring that Israeli intransigence was damaging US standing in the region, and that Arab leaders now deemed the US too weak to stand up to its Israeli ally.

Just yesterday, in testimony before the Senate armed services committee, General David Petraeus, the commander of Centcom, echoed that message, arguing that the Israeli-Palestinian conflict posed a threat to America's interests, that it "foments anti-American sentiment, due to a perception of US favoritism for Israel", and that "anger over the Palestinian question" aided al-Qaida and other jihadist groups in their efforts to recruit support. Such views have long been conventional wisdom among liberal critics of Israel, but to hear such talk out loud from America's most senior soldier in the field is breathtaking. Perhaps Obama has taken the Centcom warnings to heart and is trying to make amends.

He may have a darker purpose, hoping to notch up the first regime change of his presidency – by toppling Netanyahu. That is not wholly fanciful. The last time the US put such a serious squeeze on Israel was nearly 20 years ago, when the first George Bush threatened to withhold $10bn in loan guarantees to Israel if settlement building did not stop. That led eventually to the removal of the stubborn Yitzhak Shamir as prime minister and his replacement by the peace-seeking Yitzhak Rabin. Obama may be calculating that the same dynamic still holds today – that Israeli public opinion values good relations with the US above all else, and will cast aside any leader who gets on the wrong side of Washington.

If that is the plan, then the three demands laid down by Hillary to Bibi – the cancellation of the Ramat Shlomo construction; a confidence-building gesture towards the Palestinians; and talks on core, rather than technical, issues – are well aimed. Just the first of those could bring down Bibi's coalition. Were he to surrender the principle that Israel is allowed to build where it likes in Jerusalem, his rightwing allies would bolt. Inside the Likud, Netanyahu – who has always insisted that Jerusalem must for ever remain undivided and under Israeli control – would be discredited.

But the third demand is most intriguing. Instead of using the Biden flap simply to assert his own macho credentials – "Don't cross me" – Obama might be grasping hold of it as a rare chance to revive the near-dead peace process.

His critics say that Israel never responds to pressure, that it only makes compromises when it feels secure. Only with the full backing of George W Bush did Ariel Sharon feel able to disengage from Gaza. But the Bush Sr experience tells a different story: that it was US pressure which dragged Israel to the peace table in Madrid in 1991, spawning the Oslo accords two years later.

With that precedent in mind, Obama could dispense with the endless talks about talks that were about to get under way, shifting the ground away from process and procedure – the terrain on which Bibi is most comfortable – and to substance instead. He should do it, demanding both sides – Israeli and Palestinian – present their vision of the endgame, their statement of how they finally see this conflict being resolved. It would have to cover everything, even the most difficult areas: borders, refugees, Jerusalem. Netanyahu always says he is serious about peace. This exercise would force him, and his Palestinian counterparts, to say how serious.

Then Obama should set out a vision of his own, including "bridging proposals" to close the gap between the two sides. If he is going to spend political capital on a peace process, let it at least be spent on peace – not process. Last week Israel slapped the US vice-president in the face. Now there is a chance to make that the slap heard around the world – the one that jolts America awake.

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