An influential Saudi former military commander on making common cause with Israel and warming toward Russia as the U.S. backs away.

President Obama knew how to soothe Arab nerves rankled by his nuclear diplomacy with Iran. In May he convened a Camp David summit with the leaders of the Gulf Cooperation Council. The only problem: Of six GCC heads of state, only two showed up. The most powerful and influential, Saudi Arabia’s King Salman, wasn’t among the attendees.

The snub was a rare public expression of the kingdom’s anger at Mr. Obama. Behind Riyadh’s ire is the sense that, in its pursuit of a nuclear accommodation with Tehran, America is tilting away from its traditional Middle East allies and toward Iran’s ayatollahs. For these Arab states, the new Washington dispensation means forging security arrangements that a few years ago would have seemed unthinkable. Perhaps the most astonishing of these developments is the nascent alliance between Saudi Arabia and Israel.

Anwar Eshki, a retired major general in the Saudi armed forces, has spearheaded Riyadh’s outreach to Jerusalem. He made history in June when he appeared on a panel in Washington, D.C., with Dore Gold, the newly appointed director-general of Israel’s Foreign Ministry. At that event, Gen. Eshki outlined a vision for the Middle East that included Arab-Israeli peace, regime change in Tehran, democracy in the Arab world and the creation of a Kurdish state. And while Gen. Eshki says his outreach to the Israelis is a purely private enterprise, it hasn’t been interpreted that way in the region, in large part because he is a prominent and well-connected figure in the Saudi security establishment.

I sat down for an interview with Gen. Eshki Wednesday evening at the Prague Marriott, where he was attending a security conference.

“I believe this is a good deal, but—” he says, referring to the nuclear deal with Iran, then veering into what sounds like a carefully neutral discussion of the debate over the agreement in the U.S. A military man with a subtle and disciplined mind, he never explicitly criticizes the nuclear talks or the White House, toeing his government’s public line of tepid, conditional support for the accord. He goes so far as to dissociate himself from the anti-deal views of his one-time boss, former intelligence chief Prince Bandar bin Sultan, whom the general served as a national-security adviser when the prince was Saudi ambassador to the U.S. in the 1980s.

And yet the key is to unpack that “but.”

Throughout the talks, Secretary of State John Kerry and Iranian Foreign Minister Javad Zarif “agreed on many things,” Gen. Eshki says. “That surprised Russia, and it surprised many others.” The general also was surprised by Mr. Kerry: “He supported the Iranians!” Mr. Kerry and his boss were willing to see things Iran’s way, Gen. Eshki says, because they believe that putative moderates like Mr. Zarif can outmaneuver hard-liners like Supreme Leader Ali Khamenei and his Revolutionary Guards.

Does Gen. Eshki share that view? “I believe Iran will not change its mind as long as that regime is in power in Tehran,” he says. “Iran does many things that are not good. They want to revive the Persian Empire. And also they want to dominate the Middle East through destabilization.” Saudi Arabia is as vulnerable as Israel to such designs, if not more so, and it was the common Iranian threat that brought the general and Mr. Gold into a yearlong strategic dialogue that culminated in the Washington meeting.

“The main project between me and Dore Gold is to bring peace between Arab countries and Israel,” he says. “This is personal, but my government knows about the project. My government isn’t against it, because we need peace. For that reason, I found Dore Gold. He likes his country. I like my country. We need to profit from each other.” Jerusalem and Riyadh, he says, are two powers that “don’t want trouble in the region.”

Initially, the focus was on the Palestinian question. “We didn’t talk much about Iran at first,” Gen. Eshki recalls, “but I found that our idea and their idea was close together against Iran. We don’t like Iran to destabilize the area. We don’t like for Iran to attack Israel and destroy Israel. And we also don’t like for Israel to attack Iran and destroy Iran. This is my idea. He has another idea. But we are together.”

For Israel, the immediate Iranian-caused headache is Hezbollah, the Lebanese-Shiite terror outfit that points tens of thousands of missiles at the Jewish state. Gen. Eshki recalls once asking Mr. Gold, “ ‘When you attack Hezbollah, does Iran interfere?’ He answered, ‘No.’ ” A follow-up: “If you attack Iran, will Hezbollah support Iran?” The Israeli answered: “Yes.” Gen. Eshki’s conclusion: “Israel is thinking first of all to destroy Hezbollah, to solve the problem with Hezbollah. After that they can attack Iran.”

For the Saudis, the Iranian-backed Houthi militia in Yemen poses the immediate threat. Situated at the southwestern tip of the Arabian Peninsula, Yemen is a strategic gateway to Africa and a perennial target of Iranian meddling and al Qaeda terrorism. Gen. Eshki worked closely on Yemen issues during the 18 years he spent as a national-security adviser to the kingdom’s Council of Ministers following his stint in Washington. “I know exactly the Yemeni people, tribes and the situation,” he says. “Yemen is a central challenge for Saudi Arabia in the future.”

When the Houthis overran Yemen in the spring, Riyadh finally took action, launching a joint Arab campaign with short notice to Washington. Known as Decisive Storm, the campaign has, in fits and starts, punished and pushed back Tehran’s proxies on the kingdom’s doorstep. “Now the Storm in Yemen gave a lesson to Hezbollah and all the other [proxies] of Iran that Iran is a paper tiger,” Gen. Eshki says. “They couldn’t support the Houthis in Yemen. They couldn’t bring one plane to Yemen. For that reason, the Houthis now are talking bad against Iran on social media.”

The Islamic Republic’s imperial ambitions in the region will ultimately sound its own death knell, the general thinks. “I told the Iranians when I was there,” Gen. Eshki says. “I told [Deputy Foreign Minister Amir Hossein] Abdollahian: ‘Iran will destroy itself. If you try to revive empire, many other nationalities will ask for independence, like Azeris, like Arabs, like Turkmen, like the Baluch, like the Kurds.’ ” In other words, two can play at Tehran’s game of riling up ethnic and sectarian minorities.

Riyadh isn’t limiting itself to Jerusalem in courting potential new friends. He suggests that a thaw in the kingdom’s relations with Vladimir Putin’s Russia is under way following the rupture caused by Moscow’s sharp support for its clients, Tehran and the Assad regime, in the Syrian civil war. Riyadh and the Kremlin may now work together to stabilize Syria.

“We have to concentrate to solve the problem” in Syria, the general says. “But we don’t like Assad to stay. Because the people in Syria don’t want him to stay.” He notes that Saudi King Abdullah, who died in January, “at the beginning of the revolution called on Assad six times to solve the problem quickly: ‘Don’t kill your people. Don’t ally yourself with Iran. We need Syria united and independent.’ At the end of that, President Assad said: ‘The situation is not under my control.’ That means: Iran has much influence over him.”

Now the Kremlin is gradually coming around to Riyadh’s view of the conflict. “Russia is a great country,” he says, “but they don’t like to change their promises” to allies—in contrast to you-know-who. “Russia supported by weapons Iran and Assad in the civil war in Syria. But now Russia believes, has been convinced, that they are not in the right path. Saudi Arabia needs Russia in the Middle East, not to destabilize countries but to be a friend.”

A political solution would preserve the Syrian state apparatus while replacing the regime sitting atop it. “We don’t like that regime,” Gen. Eshki says. “There’s difference between the system and the regime. When the United States came to Iraq, they destroyed the system, and the problems ensued. We have to maintain the system but remove the regime.” He believes stabilizing the region will require a “Marshall-style project to rebuild” Syria and Yemen, a cause he personally promotes.

Such a project is the only permanent antidote to the Islamist extremism of groups like Islamic State. Using the Arabic term for the group, Daesh, the general says that its terrorism wouldn’t be possible in a country “if that country is not destabilized, if it has equity. When Syria became destabilized, Daesh came to Syria. When the government in Iraq had so much corruption and pushed the Sunni out, Daesh came to Iraq quickly. If Iraq became stabilized and strong, Daesh wouldn’t be in Iraq or in Syria.”

And contrary to fashion, Gen. Eshki still talks about democratizing the Middle East. “We have in the Gulf many problems,” he says. “We need more reform. We need more democracy in that place,” albeit democracy inflected by Islamic law. “We can’t conquer the terrorists just with weapons and security acts, but also by justice inside of the country.” He even imagines a federal future for the Arab states of the Persian Gulf region inspired by the U.S. Constitution.

He adds: “I believe Daesh will like Pac Man eat all the terrorists until it becomes one big terrorist. Then we can destroy them.”

The U.S. doesn’t figure much in the moral and strategic map Gen. Eshki paints of the region. Yes, America and Saudi Arabia are still strong allies, he says, but “the United States is trying to move from the Middle East to the Far East and the Pacific Ocean. The United States doesn’t like anymore to be involved in the Middle East, but to support the Middle East.” That may be preferable to many American voters, but it comes at the price of a diminished capacity to shape events and outcomes. Little equity, little say.

So how would the birth of a nuclear-armed Iranian theocracy affect Saudi Arabia’s new strategic path? Gen. Eshki doesn’t seem worried, but others might be: “If Iran tries to make that atomic bomb, we would do that also.”

Mr. Ahmari is a Journal editorial writer based in London.