Theresa May has “ample reasons” to cancel Donald Trump’s visit to the UK next month, Jeremy Corbyn has said.

The Labour leader said the escalating trade dispute between US and the EU, as well as the US treatment of child migrant detainees separated from their parents, were reasons enough to rescind the invitation to the US president.

His comments came as Trump threatened to impose tariffs on European cars after Brussels confirmed it would introduce retaliatory levies on American products including bourbon whiskey, Levi’s jeans and Harley-Davidson motorbikes as a response to the US steel and aluminium tariffs.

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“Donald Trump needs to understand that declaring a trade war by putting tariffs on aluminium and steel and then building up a series of tit-for-tat series of retaliations, in which the US retreats in on itself and starts a trade war with the rest of the world, is going in a very bad and dangerous direction,” Corbyn said.

Facebook Twitter Pinterest Jeremy Corbyn on his visit to the Zaatari Syrian refugee camp, in Mafraq, Jordan on Friday. Photograph: Nasser Nasser/AP

Corbyn, who was visiting a refugee camp in Jordan, told Sky News that Trump’s visit on 13 July would achieve little.

“Interestingly, it’s been downgraded time after time from a state visit to an official visit to a visit,” he said. “So I’m not absolutely sure what he’s going to do on this very brief time he has in the UK.”

“I wouldn’t have invited him and I think the prime minister’s got ample reasons to withhold the invitation if she wants to. We need to say very clearly to Donald Trump ‘We live in a multicultural society, we’re proud of it. Get over it and start living in one yourself’.”

Asked if Corbyn would invite Trump for a state visit were he in No 10, Corbyn said: “We’d have to change the date to a long way down the line.”

Theresa May offered the US president a state visit to the UK during her visit to Washington shortly after his election. However, his visit this summer has been described as a “working visit” and he is not expected to visit key areas of central London to avoid mass protests.

Instead, Trump is likely to meet the prime minister in her country retreat at Chequers and the Queen at Windsor Castle, with a visit to his Scottish golf course also a possibility.

Trump could return for a full state visit next summer. Britain’s national security advisor, Sir Mark Sedwill, was recorded suggesting the date in a Channel 4 documentary to mark the opening of the new US embassy in south-west London.