She was speaking just a day after meeting the new President in Washington, where the pair pledged their commitment to the “special relationship” between Britain and the US.
After agreeing a controversial £100 million fighter jet deal amid wide-ranging purges and security crackdowns following an attempted coup against President Recep Tayyip Erdogan, Ms May held a joint press conference with Prime Minister Binali Yıldırım.
Their talks were overshadowed by global debate over Mr Trump’s executive order to ban Syrian refugees from entering the US indefinitely, halt all other asylum admissions for 120 days and suspend travel visas for citizens of “countries of particular concern”, including Syria, Iraq and other Muslim-majority nations.
Faisal Islam, the political editor of Sky News, asked Ms May whether she viewed it as an “action of the leader of the free world”.
The Prime Minister replied that she was “very pleased” to have met Mr Trump in Washington, before evading the question by hailing Turkey’s reception of millions of refugees and Britain’s support for its government and other nations surrounding Syria.
When pressed for a second time for her view by another British journalist, Ms May continued: “The United States is responsible for the United States’ policy on refugees, the United Kingdom is responsible for the United Kingdom’s policy on refugees.”
Yvette Cooper, the former shadow Home Secretary, sent a letter to the Prime Minister urging her to echo condemnation from French and German ministers over the “deeply troubling” executive order.
Ed Miliband, the former Labour leader, said the Prime Minister's refusal to condemn Mr Trump's Muslim ban “is shocking, wrong and cannot stand”.
He added: “It flies in the face of the values of people across Britain.”
Mr Yıldırım was more direct, calling the crisis a global issue and saying that UN members “cannot turn a blind eye to this issue and settle it by constructing walls”.
“Nobody leaves their homes for nothing, they came here to save their lives and our doors were open…and we would do it again,” he added. “If there is someone in need, you need to give them a helping hand to make sure they survive.”
The Turkish Prime Minister brushed aside a question about wide-ranging human rights abuses in Turkey following the summer’s attempted coup, seeing thousands of people arrested or forced to leave government and military posts amid reports of torture in state prisons.
Ms May also evaded questions on her ally’s alleged abuses in the press conference, but had earlier urged Mr Erdogan to “sustain democracy by maintaining the rule of law and upholding its international human rights obligations”.
She and Mr Erdogan also discussed counter-terrorism, security, trade and migration in talks which stretched for an hour longer than scheduled.
Turkey has seen a succession of terror attacks by both Isis and Kurdish extremist groups since the start of the Syrian civil war, which caused millions of refugees to cross its border.
The US was among the countries resettling families from designated camps but Mr Trump has suspended all refugee admissions to the US for 120 days as part of measures he claimed would “keep radical Islamic terrorists out of the US”.
His order additionally banned Syrian refugees indefinitely until “significant changes” are made, and halved the annual cap on refugees to 50,000.
The President's executive order also suspends travel visas for anyone from seven predominantly Muslim countries, including Syria, Iraq, Libya and Iran, from entering the US for at least 90 days.
He claimed his administration needed time to develop more stringent screening processes for refugees, immigrants and visitors.
It provoked outcry from the United Nations and NGOs working to stem the worst ever global refugee crisis, with more than 65 million people forced to flee their homes.
Amnesty International warned the move could have “catastrophic consequences”, saying some of the worst fears about a Trump presidency were already being realised.
Salil Shetty, the group’s secretary general, said: “These men, women and children are the victims of the same terror President Trump claims he wants to fight against. The irony beggars belief.”
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