Barack Obama has insisted that solidarity between the US and Europe will not be affected by Brexit, as leaders flew to Warsaw for a Nato summit that the US president said was the “most important moment” for the alliance since the end of the cold war.

Speaking in the Polish capital after a meeting with EU leaders, Obama described claims that the west had been fatally weakened by Britain’s referendum vote as “hyperbole”. He emphasised the enduring strength of Washington’s relations with the EU, which he called “one of the greatest economic and political achievements of modern times.”

“This is an achievement that has to be preserved,” he adding, describing an integrated Europe as “cornerstone of US relations with the world”.

In his remarks on Friday, he did not refer directly to Washington’s ties with London, but in a Financial Times column he wrote that “the special relationship between the US and the UK will endure”.

Although he sought to play down the impact of Brexit on western cohesion, Obama described called the two-day Nato summit – in the shadow of the UK vote and in the face of a resurgent Russia and rising terrorism – as possibly “the most important moment for our transatlantic alliance since the end of the cold war”.

Obama said the lesson of the Brexit vote was that “governments cannot be remote institutions” but had to respond quickly to the fears and needs of the people.

Speaking alongside him, Donald Tusk, the president of the European Council, said that regrettable as the Brexit vote was, it was “an incident” and not the beginning of a process of European unravelling.

The Warsaw summit is focused largely on bolstering western solidarity reassuring eastern Europe against fears of Russian encroachment. Nato leaders will authorise the stationing of four multinational combat battalions in Poland, Estonia, Latvia and Lithuania, and is expected to announce a new stage of readiness of a missile defence shield in eastern Europe. Alliance officials are presenting the creation of the 1,000-strong battalions, led by the US, the UK, Canada and Germany, as a carefully calibrated response to Moscow’s more assertive military stance and intervention in Ukraine.

They insist that the defence system is intended to counter a missile threat from Iran and Syria, not to blunt Russia’s deterrent. But analysts warn that there is a risk of Russia overreacting to Nato’s moves, fuelling escalation on the latter’s tense eastern border.

The summit, is being held in Poland’s national stadium on the east bank of the Vistula river. In this location, during the summer of 1944, Stalin halted the advance of the Soviet army long enough for the Nazis to obliterate a mass uprising by the Polish resistance in Warsaw.

The Polish government has declared a high state of alert, with 18 presidents and 21 heads of government due to converge on the capital. For the duration of the summit, which will involve foreign ministers on Friday and heads of state and government on Saturday, boats will be banned along a section of the river near the stadium and light aircraft will be prohibited from flying through the country’s airspace within a 62 mile (100km) radius.

On the eve of the summit, Poland’s president, Andrzej Duda, and the Nato secretary general, Jens Stoltenberg, insisted that Brexit would not affect the strength of the alliance.

“The EU and Nato are quite separate organisations,” Duda told journalists. “The UK is one of the strongest members of Nato, and I have no doubt that its participation and cooperation in the alliance will continue at least at the same level.”

However, a Nato official said the issue was driving anxious conversations behind the scenes ahead of the summit.

“How can it not affect western cohesion? How can trillions being wiped out in market value not affect perceptions of western strength?” the official asked. He predicted that David Cameron, having resigned as prime minister, would cut a much diminished figure at the summit.

Krzysztof Blusz, a strategic analyst at the WiseEuropa thinktank in Warsaw, said the outcome of the EU referendum would have multiple indirect effects on Nato, almost all of them negative.

“Regardless [of] how and when Brexit happens, it will create a situation where Nato member states are more inward looking, more preoccupied with their nationalist tendencies. The impulse is already there, but this will add to the distraction and the Kremlin will not miss the chance to use the situation for its advantage,” he said.

Bogdan Klich, a leading opposition senator and former Polish defence minister, said he expected the UK to emphasise its Nato role to make up for voting to leave the EU, but that he worried about the wider economic impact.

“We are concerned in such a context that the EU will not have the same influence on the international arena,” Klich told the Guardian. To compensate, he argued, the EU should strengthen its defence role.

The German and French foreign ministers, Frank-Walter Steinmeier and Jean-Marc Ayrault, made the same argument in a nine-page joint policy document on post-Brexit contingencies titled “A strong Europe in a world of uncertainties”.

The document annoyed the US and Canada, which are wary of Europe duplicating Nato military structures.

Steinmeier had already caused consternation among his Nato counterparts by cautioning against “loud sabre-rattling and shrill war cries” directed at Moscow, referring to the latest military deployments on the alliance’s eastern flank as “symbolic tank parades”.

Lukasz Kulesa, a research director at the European Leadership Network, predicted that Russia’s reaction to the summit would most probably be symbolic, repackaging long-planned military upgrades as a “robust response”.

“However, things can get much worse, as Russia can make two mistakes while interpreting the results of Warsaw. The first mistake would be to read too much into the summit outcome and assume that the alliance is gearing up for a confrontation,” Kulesa said. The second possible mistake is that Moscow could take it too lightly “and conclude that Nato’s unity on Russia and deterrence resolve is rather shallow, and may be broken if more pressure is applied”. Either miscalculation could be destabilising, he argued.

Klich said: “I don’t have any doubt that Russia will use this little military presence of the US and other Nato states as one of its arguments to raise the level of tension with the alliance.” However, he argued that if Nato did not strengthen its eastern flank in response to Russian actions in Ukraine, “it would be a puppet, not a tiger”.