Donald Trump’s insults towards Keir Starmer are “small and petty” and designed to put pressure on the prime minister to change his position on Iran, David Lammy has said, as he insisted the UK would not get dragged into the conflict.

The deputy prime minister argued the US president should be able to “disagree agreeably” with allies rather than publishing attacks on social media, and that US actions had “made things worse, not better” as far as global instability was concerned.

The UK’s relationship with Washington has become increasingly strained as the conflict continues, with Starmer heavily criticised and at times mocked by Trump for not committing British equipment and personnel to fighting the war.

In response, Starmer has said he is “fed up” with the effect that Trump’s actions in the Middle East are having on British energy bills, while Rachel Reeves, the chancellor, has described the conflict as a “mistake” that is wreaking havoc on the global economy.

In an attempt to repair relations and stress the urgency of reopening the strait of Hormuz, a crucial shipping lane through which much of the world’s oil and goods pass, Lammy flew to Washington last week to hold talks with JD Vance, the US vice-president, and the secretary of state, Marco Rubio.

In his first interview since the visit, Lammy said he had “no doubt” that all US politicians “purely through self-interest alone” understood the economic impact of the war, and that Vance himself was concerned about the cost of living in the run-up to the US midterm elections.

On Sunday, Trump announced US negotiators would head to Pakistan this week to resume peace talks, though he warned the US would “knock out” every power plant in Iran if it did not accept a deal, and the strait remains blocked.

Asked whether he felt reassured that the US was serious about ending the conflict, Lammy said: “The only issue that anyone wants to talk about is affordability and prices. This war, on top of the war in Ukraine, on top of the pandemic, has made things worse, not better. The quicker it ends, the better for everybody.”

Speaking at a conference of global progressives in Barcelona, Lammy said: “Donald Trump has tried to put pressure on the prime minister to change his position on Iran, and he isn’t going to. This is not our war, and we’re not going to get engaged.

“The rhetoric is intense, but I think the public are beginning to see it for what it is. It sometimes feels petty and small and certainly I think that comments a few weeks ago about the contribution that our armed forces made in Afghanistan, for example, are just incomprehensible.”

Lammy said he had stressed to Vance the importance of reopening the strait for the sake of the global economy. “He talked about the price of petrol in the US. My sense was that he himself recognises the effect on affordability, and he is heading to midterms, and I think he’s focused on that.”

Later this month, King Charles and Queen Camilla will head to the US for a state visit, amid some concern at home that Charles could be placed in an embarrassing predicament given the strained relations.

But Lammy said: “I fully expect the state visit to be a huge success. All of the politicians that I spoke to are also able to recognise that this is also not just any old state visit. It’s a celebration of 250 years of independence.”

The visit was an important moment to reflect on the transatlantic relationship over time, at a time when it was under pressure, he added.

“Let’s be absolutely clear, Churchill and Roosevelt had disagreements over how to handle Stalin; Wilson … disagreed with the American administration at the time on Vietnam; even Margaret Thatcher and Ronald Reagan disagreed over Grenada.

“We do have a special relationship for many reasons, but notwithstanding that, we can also disagree. It is better, in my view, when you disagree agreeably, and that is what we have been traditionally used to. I lament, frankly, the social-media age in which we’re not living up to the best of standards.”

Turning to the EU, Lammy, who met left-leaning leaders including Spain’s Pedro Sánchez and Brazil’s Luiz Inácio Lula da Silva, said the UK wanted even greater alignment on goods in the run-up to the next election.

“I sense that, both in Europe and in the UK, the mood recognises that in tough economic times … it is axiomatic that greater alignment with the EU is beneficial for all of us,” he said.

“These are times where you have to huddle together, particularly in an age where we’re seeing a renewed protectionism.”

However, the committed europhile said he had “stopped doing crystal balls” about whether he could see the UK rejoining the EU in his lifetime.

“I came into politics in an age where there was a centre ground in politics. I never dreamed that I would see the kind of populism and some of the extreme nasty rhetoric – particularly the nativist, racial undertones – that you now see in public life.

“So I stopped predicting. But what I do have faith in is the UK to act in its self-interest in tough economic times, and its self-interest is clearly greater and closer proximity to our biggest trading partner, and that is the EU.”

In the last week, Lammy has travelled to the US, the Gulf and Spain on behalf of the prime minister – and is regularly consulted by him on US relations in particular.

When asked about the suggestion he was the “real” foreign secretary, he laughed and said he had just spoken with Yvette Cooper, after her diplomatic tour of the Middle East. “She’s on the day to day, and I can assist where I can assist, but she is absolutely driving here,” he said.

Lammy said he felt that Starmer’s domestic position had been bolstered by his handling of the conflict but acknowledged next month’s local elections would be “tough” and some voters would give Labour a bloody nose. But he suggested it was still possible for ruling parties to “recalibrate”.