A day of whirlwind diplomacy for Donald Trump on Saturday, including calls to five world leaders such as the Russian president, Vladimir Putin, was overshadowed by a global backlash against his ban on refugees.

The “congratulatory call” with Putin lasted an hour, the White House said in a short statement, and ranged from discussion of “mutual cooperation” to defeat the terror group Isis to negotiating an end to the Syrian civil war.

“The positive call was a significant start to improving the relationship between the United States and Russia that is in need of repair,” the White House said.

In the Kremlin’s more detailed account, Trump and Putin discussed “partnership” on a wide range of international issues, including wars in the Middle East, the Israeli-Palestinian conflict, Iran’s nuclear programme, the Korean peninsula and the simmering war in Ukraine, where Russia has supported separatist rebels since 2014.

Cooperation in Ukraine would represent a stark turn in US policy, which has supported Kiev and imposed punitive sanctions against prominent Russians for the Kremlin’s role in the crisis.

Those sanctions, which Trump has mused about removing, were not mentioned in the Russian readout of the call, but the Kremlin said the leaders had discussed “the importance of restoring mutually beneficial economic-trade links”.

The Kremlin also stressed possible coordination in fighting terrorism in Syria, where Putin has supported the dictator Bashar al-Assad with a huge bombing campaign.

In unusually effusive language, the Kremlin said Trump and Putin plan to meet in person, and that they had kind words for each other’s nations.

“Donald Trump asked to convey wishes of happiness and prosperity to the Russian people, mentioning that the American people hold Russia and her citizens in sympathetic regard,” the Kremlin said.

“Vladimir Putin, for his part, emphasized that in Russia there are analogous feelings toward Americans. He recalled that for the span of two centuries our nation had supported America, been allies in two world wars, and now views the US as its most important partner in the fight against international terrorism.”

Trump also spoke to the German chancellor, Angela Merkel, and French president, François Hollande, both of whom are sharply at odds with his radical refugee vetting policy.

The White House said Merkel and Trump spoke about Russia and Ukraine, and “agreed on the Nato alliance’s fundamental importance to the broader transatlantic relationship and its role in ensuring the peace and stability of our North Atlantic community”.

Merkel has stood staunchly against Trump’s denigrations of Nato, which he has called “obsolete”. He has also accused allies of failing to pay their dues for military defense, a dispute alluded to by the White House, which said the leaders discussed “appropriate investment in military capabilities to ensure all Allies are contributing their fair share to our collective security”.

The White House did not say whether Merkel and Trump discussed refugees, hundreds of thousands of whom Germany has received in the last year. In a recent interview, Trump described Merkel’s open-door policy as “a catastrophic mistake”.

Trump has had a rollercoaster ride in his first week as a world statesman. His meeting with the British prime minister, Theresa May, at the White House passed successfully but a crisis erupted with neighbouring Mexico, whose president, Enrique Peña Nieto, abruptly cancelled a planned visit.

Trump had moved ahead with plans for a border wall and appeared to threaten a hefty border tax on Mexican imports. The spat played out on Twitter.

The US president began Saturday by calling the Japanese prime minister, Shinzo Abe, who visited him at Trump Tower in New York during the transition.

“President Trump affirmed the ironclad US commitment to ensuring the security of Japan,” the White House said. During the election campaign, Trump suggested that America’s defence of Japan was too one-sided and expensive.

Trump and Abe also discussed the threat posed by North Korea and “committed to deepen the bilateral trade and investment relationship” ahead of a meeting in Washington on 10 February, the White House added. Earlier this week, the US withdrew from the Trans-Pacific Partnership, a flagship 12-country deal drawn up by Barack Obama, raising the prospect of a bilateral agreement with Japan.

Earlier on Saturday, the French foreign minister, Jean-Marc Ayrault, held talks with Germany’s Sigmar Gabriel in Paris. Ayrault said Trump’s order on Friday – barring all refugees from entering the US for four months and those from war-ravaged Syria indefinitely – “can only worry us”.

The French minister told a press conference: “We have signed international obligations, so welcoming refugees fleeing war and oppression forms part of our duties.”

Gabriel told reporters: “The United States is a country where Christian traditions have an important meaning. Loving your neighbour is a major Christian value, and that includes helping people. I think that is what unites us in the west, and I think that is what we want to make clear to the Americans.”

Trump’s executive order, which he said will keep out “radical Islamic terrorists”, has been condemned within the US and around the world. No visas will be issued for migrants or visitors from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen for at least 90 days.

The Iranian president, Hassan Rouhani, said on Saturday now was “not the time to build walls between nations”. Speaking at a tourism convention in Tehran, he added: “They have forgotten that the Berlin Wall collapsed many years ago. Even if there are walls between nations, they must be removed.”

An Iranian foreign ministry statement said the executive order was an “open affront against the Muslim world and the Iranian nation” and said Iran would “take appropriate consular, legal and political measures” in response.

“Instead of countering terrorism and protecting American people, these measures will be written in history as a gift to extremists and their supporters,” the statement said.

The US president’s admiration for the authoritarian Russian leader has caused consternation in America and beyond. On Friday he said it was “very early” to be talking about possibly lifting sanctions that were imposed on Russia over its incursion into eastern Ukraine.

US intelligence agencies found that Russia had interfered in the presidential election with the aim of harming Hillary Clinton and helping Trump. House and Senate intelligence committees have announced investigations into possible links between Moscow and members of the Trump campaign.

Members of Trump’s own party have warned against lifting sanctions. John McCain, chairman of the Senate armed services committee, said he would try to “codify sanctions against Russia into law”.

In the past three years, McCain added, Russia has invaded Ukraine, annexed Crimea, threatened Nato allies, intervened militarily in Syria and interfered in the US election with cyber-attacks and a disinformation campaign.

The senator called Putin “a murderer and a thug who seeks to undermine American national security interests at every turn”, and warned Trump: “For our commander-in-chief to think otherwise would be naive and dangerous.”