Iran’s president, Hassan Rouhani, has criticised the British prime minister, David Cameron, a week after a landmark meeting between the two leaders on the sidelines of the UN general assembly.

Rouhani said it was unacceptable and wrong for Cameron to say, within hours of them meeting in New York, that Tehran was “part of the problem” in the Middle East. In his closing speech at the general assembly last Thursday, the UK prime minister said Tehran’s support for what he said were terrorist organisations had to change, although he added that Iran could help the west in its fight against Islamic State (Isis).

That meeting marked a milestone in Tehran and London’s relations, which have been strained since the 1979 Islamic revolution, but Cameron’s comments upset Tehran hardliners and embarrassed Rouhani at home.

“Iran should also be given the chance to show it can be part of the solution, not part of the problem,” Cameron said during his speech, according to a transcript published on the government’s website. “We have severe disagreements. Iran’s support for terrorist organisations, its nuclear programme, its treatment of its people. All these need to change.”

Rouhani, who was speaking to reporters after arriving in Tehran from a conference in the Russian city of Astrakhan, said the request for the meeting was made by the British prime minister. “He had asked for a meeting and we had agreed and that was how we met,” he said late on Monday.

“He made comments later at his [UN] speech which were wrong and unacceptable,” Rouhani told Iranian reporters, according to Iran’s state television. Rouhani said having different opinions over political issues was normal.

He added: “We live in the 21st century. If someone wants to go back and think in the ways of the 19th century, that’s their own loss.”

Cameron went on in his speech to say that Tehran could help the west with the Isis threat. “Iran’s leaders could help in defeating the threat from Isil [Isis]. They could help secure a more stable, inclusive Iraq, and a more stable and inclusive Syria. And if they are prepared to do this, then we should welcome their engagement.”

Cameron’s speech at the UN prompted an immediate reaction from Iran’s foreign ministry last week but Monday’s remarks were the first by Rouhani.

Following Cameron’s speech, the Iranian foreign minister’s spokeswoman, Marzieh Afkham, said many of the dilemmas in the Middle East today were the result of the policies adopted by Britain in the past.

“The speech by the British prime minister at the UN general assembly shows the perpetuation of the egocentric attitude of a government which has a history of [causing] trouble in our region,” she said on Thursday, according to Iran’s English-language television Press TV.

Given the significance of the talks, Rouhani had to weigh up every step he took as he met the British prime minister. And choice of words has huge repercussions in Tehran. Iranian hardliners are deeply sceptical of British politics and have long accused London of interfering in Tehran’s internal affairs.

In the Iranian hardliners’ view, Cameron’s comments indicated that Rouhani was wrong to meet him in New York.

“It’s not surprising to be tricked by the old fox,” the hardline news agency SNN said, referring to Britain by an old nickname used by some Iranian hardliners.

“Cameron didn’t even wait for the ice of London-Tehran ties to thaw,” SNN said.

The conservative Vatan-e-Emrooz newspaper said Cameron had insulted Iran and published a front-page depicting a picture of the meeting with the headline: “The fox kicks.”