To say that Europeans will welcome President George Bush on his farewell visit to Europe next week would invite a charge of verb-abuse. Welcome is hardly the word. But they will be glad to see the back of him. His two terms have been a bad time for relations between Europe and the United States. The question now is how much better those relations will get under a President Barack Obama or John McCain. My guess is: with Obama, it will be very different and an exciting ride, but still not easy; with McCain, a little better than with Bush at the outset, but could rapidly get stormy again.

What we're asking here is actually a deeper question: how much does the individual matter in history? Answer: a lot. If the winner of the 2000 presidential election had been Al Gore (ie the winner of the 2000 election) the story of transatlantic relations over the past few years could have been very different. The 9/11 attacks might have provoked a trans-atlantic crisis anyway, because America then felt itself to be at war while Europe didn't. But so much of the subsequent bust-up had to do with Bush himself: his unilateralism, his obsession with Iraq, his cowboy style, his incompetence.

There has been some improvement in transatlantic relations during his second term. The Spenglerian doom-and-gloom prophecies of five years ago, at the height of the Iraq crisis, look a bit comical now. The coming "clash of civilisations", predicted one American foreign policy expert, would not be between the west and Islam but between Europe and America. "Americans are from Mars and Europeans are from Venus," wrote the American neoconservative Robert Kagan. It turns out we're from and on the same planet, after all. As European cities like Madrid and London have been hit by takfiri jihadist terrorism, as even conservative Americans have recognised that you can't win your "war on terror" like a conventional war, so there has been some convergence on what this long struggle involves.

Yet a few big, awkward facts remain. During the cold war, the transatlantic west was held together by a common enemy. No longer. For all that transatlantic convergence on the threat of international terrorism, it doesn't pull us together like the Soviet threat did. ("If only we had Brezhnev back," sighed a former British foreign secretary at the height of the Iraq crisis.) A recent comparison of British and American counter-terrorist strategies in the New York Review of Books makes the point that for Britain - and much of Europe - terrorism is an enemy within, like cancer, whereas for most Americans it's still an enemy without. What's more, analysts in Washington regard Europe itself as a threat to the national security of the United States, because the old continent is now home to potential jihadist terrorists.

In the bipolar world of the cold war, western Europe and America were condemned to work together. In today's multipolar world, there are more possible permutations. The US has a burgeoning love affair with India. It may prefer such large, friendly non-western democracies to our carping little old European ones. Europe's energy dependency on Russia, and its growing economic dependency on China, may tempt European countries to cuddle up to those authoritarian giants more than Washington would like. In this brave new world, there's nothing inevitable about the transatlantic alliance.

Because the structural ties that bind are weaker, the personalities, visions and strategies of leaders on both sides of the pond are more important than ever. I'll leave Obama for another week, and concentrate on the more difficult case. McCain's youthful biography commands respect, like that of the older Bush and unlike that of the younger Bush. But he is an old man now and he holds none of Obama's fascination for Europeans. If "soft power" means "the power to attract", then Obama is the personification of American soft power. McCain is not. Moreover, he has a famously volcanic temper - not necessarily an asset when dealing with time-consuming and self-important European leaders.

Quips about "McBush" and "McSame" may be too simple. McCain does send some welcome new messages: renounce torture, close Guantánamo, practise "international good citizenship" on climate change. But he also has a lot of foreign policy ground in common with his predecessor. In a speech in Los Angeles earlier this year, he made a big point of how his personal experience of combat has led him to "detest war", but the fact is that he operates, at least as much as Bush, within the paradigm or metaphor of "the nation at war". What many people see and like in him is precisely the image of the warrior chief. He's the guy who said America could still win in Iraq, when all about him were giving up.

"Defeating radical Islamist extremists is the national security challenge of our time," he wrote in Foreign Affairs last year. "Iraq is this war's central front, according to our commander there, and according to our enemies, including al-Qaida's leadership." And according to George Bush. But not according to most of America's military, security and intelligence experts, friends around the world and European allies, all of whom would reply: (a) that there is no "central front"; and (b) that in this struggle Afghanistan, Pakistan, Saudi Arabia and the Muslim communities of Europe are at least as important as Iraq.

He has talked of "rogue-state roll-back" and speaks a language of confrontation with Iran. Since Iran is pressing ahead with building up its uranium enrichment capacity at an alarming rate, sometime in the next four years he could face the decision whether to bomb its nuclear facilities. For relations between the US and Europe, Iran could then become another Iraq - only worse.

Describing himself as "a realistic idealist", McCain takes counsel both from the neocons who gained the upper hand after 9/11 in the administration of the younger Bush and the foreign policy realists who prevailed both before and after Europe's 9/11 (November 9 1989, the fall of the Berlin Wall) in that of the older Bush. On the neocon idealist side, he takes from Robert Kagan the idea of a league of democracies. He even suggests that Russia should be kicked out of the G8 to make room for Brazil and India. And McCain, like Bush in his second term, endorses a strategy of democracy promotion in the wider Middle East.

We Europeans - and Canadians, Australians, Indians and other small "d" democrats around the word - should not be scared by any of this, but we should be prepared. Kagan rightly points out that the Danish prime minister, Anders Fogh Rasmussen (spoken of as a possible president of the European council) has himself suggested an "alliance of democracies". We need to have our answers ready, by November, to the proposals likely to come lobbing over the Atlantic: here's where we agree, this is how we'd do things differently. We need our own post-Bush to-do list for a revived transatlantic partnership. And we can not count on Obama winning. After the rough ride with Bush, prepare for a rough ride.

timothygartonash.com