U.S. Secretary of State Rex Tillerson took the Trump administration's case for isolating and containing Iran in the Middle East and beyond to two Gulf Arab nations on Sunday, pushing for Saudi Arabia and Iraq to unite to counter growing Iranian assertiveness.

He also called for a quick resolution to the ongoing crisis between Qatar and its Arab neighbours, which he said was unintentionally bolstering Iran.

In Saudi Arabia and later Qatar, Tillerson denounced Iran's "malign behaviour" and urged nations of the region and elsewhere, notably Europe, to join the administration to halt any business they do with Iran's powerful Revolutionary Guard Corps.

'Those fighters need to go home'

He also demanded that Iranian and Iran-backed Shia militia in Iraq either return to their homes, integrate into the Iraqi army or leave the country.

"Those fighters need to go home," Tillerson said. "Any foreign fighters need to go home."

In Riyadh for the inaugural meeting of the Saudi Arabia-Iraq Coordination Council — a vehicle that U.S. officials believe can wean Iraq from Iran — Tillerson told Saudi King Salman and Iraqi Prime Minister Haider al-Abadi that the nascent partnership between their countries held great promise for Iraq's reconstruction after devastating battles to wrest territory from Islamic State in Iraq and Syria (ISIS) and its independence from foreign influence.

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Al-Abadi, right, among leaders in Riyadh for the inaugural meeting of the bilateral co-ordination council, expressed pleasure with 'the thriving relations' between Iraq and the U.S. (Alex Brandon/Associated Press)

"We believe this will in some ways counter some of the unproductive influences of Iran inside of Iraq," he said at a news conference with Saudi Foreign Minister Adel al-Jubeir after the council meeting.

Shunning Revolutionary Guards

Tillerson said countries outside the region could also play a role, primarily by shunning the Revolutionary Guards, which play a major role in Iran's economy and were added to a U.S. terrorism blacklist earlier this month. Companies and countries that do business with the guards "really do so at great risk," he said.

"We are hoping that European companies, countries and others around the world will join the U.S. as we put in place a sanctions structure to prohibit certain activities of the Iranian Revolutionary Guard that foment instability in the region and create destruction in the region," Tillerson said.

At the council meeting, Tillerson praised the Saudi king and al-Abadi for the August reopening of a major border crossing and the resumption of direct flights between Riyadh and Baghdad last week.

"Both represent the beginning of what we hope will be a series of even more tangible actions to improve relations and strengthen co-operation on a host of issues," he said. "Your growing relationship between the kingdom and Iraq is vital to bolstering our collective security and prosperity and we take great interest in it."

His participation in the meeting comes as U.S. officials step up encouragement of a new axis that unites Saudi Arabia and Iraq as a bulwark against Iran's growing influence from the Persian Gulf to the Mediterranean Sea. Amid the push for that alliance, the Iraqi government is struggling to rebuild recently liberated Islamic State strongholds and confronts a newly assertive Kurdish independence movement.

'We are facing in our region serious challenges in the form of extremism, terrorism as well as attempts to destabilize our countries. These attempts require our full attention.' - Saudi King Salman

History, religion and lots of politics stand in Tillerson's way, but both the Saudi king and the Iraqi prime minister appeared optimistic about the prospects.

"We are facing in our region serious challenges in the form of extremism, terrorism as well as attempts to destabilize our countries," Salman said. "These attempts require our full attention. ... We reaffirm our support for the unity and stability of our brother country of Iraq."

Al-Abadi expressed pleasure with "the thriving relations between our two brotherly countries."

"We are open and we want to move away from the past," he said. "The region cannot tolerate any further divisions. Interference in the internal affairs of other state should stop."

Shia-majority Iraq and Sunni-led Saudi Arabia, estranged for decades after Saddam Hussein's 1990 invasion of Kuwait, have tried in recent years to bridge their differences. Saudi Arabia reopened its embassy in Baghdad in 2015 after a quarter century. The first visit by a Saudi foreign minister to Baghdad came in February this year, followed by the border crossing reopening in August and resumption of direct flights between the capitals suspended during the Gulf War.

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Salman, centre, shakes hands with Tillerson after the meeting. (Alex Brandon/Associated Press)

Over the weekend, the Saudi oil minister, Khalid al-Falih, made a high-profile appearance at Baghdad's International Fair, and held talks with his Iraqi counterpart, Jabar al-Luabi.

Nevertheless, the relationship is plagued by suspicion. Iran's reported intervention in Iraq's semiautonomous northern Kurdish region, after last month's much criticized vote for independence in a referendum, has deepened the unease.

Qatar crisis

After his talks in Riyadh, Tillerson flew to the Qatari capital of Doha, a direct route that has been closed to commercial airlines since June when the now five-month-old crisis between Qatar and Saudi Arabia, Bahrain, Egypt and the United Arab Emirates erupted, ostensibly over terrorism financing. Qatar and Bahrain are home to the largest U.S. military bases in the Middle East.

Tillerson has attempted to facilitate a dialogue through talks with the feuding parties as well as supporting a Kuwaiti mediation effort, but has thus far been unsuccessful. He renewed those calls but allowed that progress seemed unlikely.

"We cannot force talks between parties who are not ready to talk," he said.

"The United States remains concerned that the dispute has had negative consequences economically and militarily; the U.S. has felt these effects as well," he said. "None of us can afford to let this dispute linger. We ask that everyone ease the rhetoric and deescalate the tensions."

Tillerson noted that the only country benefiting from the crisis is Iran, which is now Qatar's lifeline as its neighbours have sealed their land, sea and air borders. He said Qatar's new reliance on Iranian airspace is "the most immediate and obvious gain that Iran has."

"Any time there is conflict and destabilization among countries that are typically allies, someone will always come in to exploit those differences," he added.