In his first White House televised interview, with the Al Arabiya news network based in Dubai, United Arab Emirates, President Obama buried the lead: The war on terror is over.

Yes, the with-us-or-against-us global struggle — the so-called Long War — in which a freedom-loving West confronts the undifferentiated forces of darkness comprising everything from Al Qaeda to elements of the Palestinian national struggle under the banner of “Islamofascism” has been terminated.

What’s left is what matters: defeating terrorist organizations. That’s not a war. It’s a strategic challenge.

The new president’s abandonment of post-9/11 Bush doctrine is a critical breakthrough. It resolves nothing but opens the way for a rapprochement with a Muslim world long cast into the “against-us” camp. Nothing good in Israel-Palestine, Afghanistan or Iran could happen with that Manichean chasm.

Obama said, “The language we use matters.” It does. He said he would be “very clear in distinguishing between organizations like Al Qaeda — that espouse violence, espouse terror and act on it — and people who may disagree with my administration and certain actions, or may have a particular viewpoint in terms of how their countries should develop. We can have legitimate disagreements but still be respectful.”

Bush liked to distinguish between terrorists and the moderate, freedom-loving Muslims of his imagination. Obama makes a much more important distinction here: between those bent on the violent destruction of America and those who merely dislike, differ from or have been disappointed by America.

These days the great majority of the world’s Muslims fall into the latter category. Obama is right to take his case to them through the Arabic-language Al Arabiya network.

His tone represented a startling departure. He was subtle, respectful, self-critical and balanced where the Bush administration had been blunt, offensive, bombastic and one-sided in its embrace of an Israel-can-do-no-wrong policy.

Speaking as his Middle East envoy, George Mitchell, began an eight-day visit to the region, Obama described the mission as one of listening “because all too often the United States starts by dictating.”

Obama went further. Citing Muslim members of his own family and his experience of life in a Muslim country (Indonesia), he repositioned the national interest and his own role.

He defined his task as convincing Muslims that “Americans are not your enemy” and persuading Americans that respect for a Muslim world is essential. His objective, he said, was to promote not only American interests but those of ordinary people — read Muslims — suffering from “poverty and a lack of opportunity.”

That’s a significant ideological leap for an American leader, from the post-cold-war doctrine of supremacy to a new doctrine of inclusiveness dictated by globalization — from “the decider” to something close to “mediator-in-chief.” I applaud this shift because it is based in realism; a changed world is susceptible to American persuasion, not to American diktat.

Still, words do not alter the fact that the post-Gaza challenge facing Obama is immense. Here in Iran, where anti-American rhetoric is too significant a pillar of the 30-year-old Islamic Revolution to be lightly sacrificed, the response to the president’s interview was cool. It came as the government, citing the Israeli assault on Gaza, approved a bill to investigate and prosecute alleged war crimes anywhere in the world.

President Mahmoud Ahmedinejad said change under Obama was good but would only be credible if America apologized to Iran for its role in the 1953 coup, among other things. The hard-line daily Kayhan said: “Obama follows Bush’s footsteps.”

In fact, Obama said he would pursue dialogue with Iran and praised the greatness of Persian civilization even as he deplored Iranian threats against Israel, its nuclear program and “support of terrorist organizations in the past.”

Any U.S.-Iranian dialogue will have to be rooted in a word Obama favors: respect. The United States has underestimated Iranian pride and the fierce attachment to its independence of a nation that has known its share of Western meddling.

Carrots and sticks will lead nowhere. Nor will an exclusive focus on the nuclear issue that fails to examine the whole range of American and Iranian interests, some shared, some hotly contested.

What is certain, with Iran as with the rest of the Middle East, is that there will be setbacks. Terrorists will attack. Obama will be denounced. But as Mitchell knows from his experience of bringing peace to Northern Ireland, the critical thing is perseverance.

Tony Blair, now also a Middle East envoy and Mitchell’s partner in Belfast, once put it to me this way: “The only reason we got the breakthrough in Northern Ireland was we did in the end focus on it with such intensity over such a period that every little thing that went wrong — and everything that could go wrong did at some point — was all the time being managed and rectified.” He described the approach as: “Any time we can’t solve it, we have to manage it, until we can start to solve it again.”

Bush had the ideological framework wrong. Obama has righted it by ending the war on terror. Now comes the hard Middle Eastern slog of solve-manage-solve. It will need the president’s unswerving focus.