Barack Obama's flying visit to Baghdad made for a fitting conclusion to this unusually long and varied presidential tour. It was a tour, moreover, on which the novice US leader was rarely less than pitch-perfect. For hopeful Europeans, as for Turkey, George Bush's problematic eight years in the White House were thoroughly laid to rest.

If it was re-engagement with America's old friends and allies that the new President was after, he was amply rewarded – but only because, in his phrase, he arrived with hand outstretched. We cannot know what took place behind closed doors, but we can judge, from the smiles of such stony-faced characters as the Russian President, Dmitry Medvedev, and China's President, Hu Jintao, that a new international atmosphere has been established.

The G20 London summit may have produced more style than substance. But style and, above all, tone are not to be despised in international relations. Getting such things right is something many new national leaders have to learn and some never really master. Mr Obama, and – it must also be said – his wife, are naturals.

For any national leader – even one who has triumphed in a marathon US presidential campaign – this would have been an exhausting tour, demanding a different approach, and different expertise, at every stop. Yet Mr Obama seemed to draw new strength from each encounter. The bigger and more engaged the audience, the more energised he seemed to be.

In London there was the economic crisis, high diplomacy and protocol. In Strasbourg there was the finely balanced French-German duo to please, and the Nato allies to be cajoled over Afghanistan. In Prague, Mr Obama met "new" Europe face to face and set out his vision for a world with far fewer, and eventually no, nuclear weapons – even as North Korea conducted its latest abortive rocket test.

He arrived in Turkey with the message that the US was not "at war with Islam" and that its relations with the Muslim world would not be defined by opposition to al-Qa'ida. In Istanbul, he pressed home his ecumenical theme by meeting leaders of all the city's main religions. And thence to Baghdad, as Commander-in-Chief, to address the servicemen whose eventual withdrawal he had announced as one of his first presidential acts.

To pull all this off and leave so few dissatisfied in his wake is a considerable feat. Not for the first time, we have to go back as far as JFK for comparisons. If Mr Obama's main objective was to cast the United States as a different type of global player – more culturally sensitive, more collegiate – then he succeeded. To demand more of a relatively young President in office not yet 100 days would be unreasonable. Yet what was, without doubt, a personal and political triumph leaves two questions.

The low-key geniality favoured by Mr Obama was of a piece with his early pledges to listen. But a time will come when listening must give way to doing, and then it will be harder to please everyone. We were certainly watching an accomplished politician and orator, but were we watching a world leader in the making?

The second question is as much about America as its president. Even before he set off, some erstwhile supporters were already voicing disappointment that he had not been more radical. Others argued that he should be devoting all his time to the US economy rather than traipsing around foreign parts. On his return, Mr Obama will learn what many a US president learned before him: acclaim abroad rarely translates into higher approval at home.