The order, which came into force as soon as Trump signed it, requires US border officials to turn away any traveller from Syria, Iraq, Iran, Sudan, Libya, Somalia and Yemen for the next 90 days.
With only a few exceptions for diplomats and dual citizens, the order takes no account of whether travellers have already been issued with visas or have visited the United States before.
There were immediate reports of travellers who had been issued visas for travel being turned away or told not to board flights because of the ban.
They included an Iranian film director nominated for an Oscar in the Best Foreign Film category who will be unable to attend this year's ceremony in the wake of President Trump's ban.
Asghar Farhadi is nominated for The Salesman, which tells of a couple whose relationship is thrown into disarray after an intruder surprises her in the shower.
Meanwhile, Mohammed Al Rawi, a graduate of California State University and former journalist with the Los Angeles Times, said his father had been hauled off a flight in Qatar as a direct result of Trump's decision.
He wrote on Facebook hours after the order was signed:
Five Iraqi passengers and one Yemeni were barred from boarding an EgyptAir flight from Cairo to New York on Saturday.
The passengers, arriving in transit to Cairo airport, were stopped and re-directed to flights headed for their home countries despite holding valid visas, Reuters reported.
Some airlines have warned that all passengers whose journeys began in any of the seven countries may be affected, even if their own citizenship is not on the "banned" list.
Vera Mironova, a Russian citizen returning from an academic research trip to Iraq, said she had been warned at check in that she may not be allowed into the US despite holding a green card.
"I just talked to Lufthansa guys and since an hour ago they need to inform all people travelling from Iraq about this possibility," she said before boarding on Saturday afternoon.
Donald Trump's decision to close America's borders to refugees was causing confusion and chaos at airports across the US, as people fleeing war zones were turned away by customs officials.
But the ban is now being met with several high profile challenges from lawyers at civil rights organisations who say that the demands made in the executive order may be illegal.
The Immigration and Nationality Act, implemented by congress in 1965 banned all discrimination against immigrants on the basis of national origin. President Lyndon B Johnson said as he signed the law that "the harsh injustice" of the national-origins quota system had been "abolished."
The law states that no person could be "discriminated against in the issuance of an immigrant visa because of the person's race, sex, nationality, place of birth or place of residence."
The detentions at airports around the US have prompted legal challenges. The New York Times reported that lawyers representing two Iraqi refugees held at Kennedy Airport filed a writ of habeas corpus early Saturday in the Eastern District of New York seeking to have their clients released. At the same time, they filed a motion for class certification, in an effort to represent all refugees and immigrants who they said were being unlawfully detained at ports of entry.
One of the Iraqis, Khalid Darweesh, worked for the US government in Iraq for a decade. Whilst the other, Haider Sameer Abdulkhaleq Alshawi, was coming to the United States to join his wife and young son, the lawyers said.
The attorneys said they were barred from meeting with their clients. When they asked who they needed to talk to to remedy this, a Customs and Border Protection official told them to "call Mr Trump".
One of the lawyers is from the Refugee Assistance Project. The group said in a statement that the executive order is "irresponsible and dangerous".
The organisation said: "Denying thousands of the most persecuted refugees the chance to reach safety is an irresponsible and dangerous move that undermines American values and imperils our foreign relations and national security.
"IRAP works with hundreds of the most vulnerable refugees - children with medical emergencies, survivors of gender-based violence and torture, and Afghan and Iraqi allies to US forces, to name a few - who will be left in immediate life-threatening danger.
"For many of them, resettlement in the United States is their only option to live safely and with dignity."
The detention of a man who served the US military is particularly abhorrent to Matt Zeller, founder of No One Left Behind, which aims to help Iraqi and Afghan people who worked for the US military secure special immigrant visas.
He said America is breaking its promise to men and women who served the US military at great personal risk to themselves - which is not only wrong, he said, but also undermines trust in the United States and endangers the lives of any future service member sent overseas.
"This is going to get future Americans killed in future wars. It comes down to that," he said. "We're never going to live down this shame if we let this go on."
An Army veteran of the war in Afghanistan, Zeller said his life was saved by his Afghan interpreter, who served the US military there for nine years. During the past 24 hours, he said he has fielded requests for help from nearly 1000 clients in Afghanistan and Iraq, most of whom have been waiting years to be processed through what Zeller called "the most extreme vetting process our country can muster."
The order also sparked concern in the business world.
Google recalled all travelling staff members to the United States following the order amid concern about the possible impact on recruiting top talent abroad.
A Google spokesperson said: "We're concerned about the impact of this order and any proposals that could impose restrictions on Googlers and their families, or that could create barriers to bringing great talent to the US. We'll continue to make our views on these issues known to leaders in Washington and elsewhere."
Mark Zuckerberg, the founder of Facebook, wrote in a post on Saturday that he was "concerned" about the impact of the orders and that he would be working with Fwd.US, a charity he supports, to develop protections for child immigrants brought to the US at a young age by their parents.
My great grandparents came from Germany, Austria and Poland. Priscilla's parents were refugees from China and Vietnam....
Posted by Mark Zuckerberg on Friday, 27 January 2017
The order signed by Trump also imposes a 120-day suspension the US refugee resettlement programme, regardless of applicants' country of origin, while administration officials develop additional vetting procedures and decide which countries those procedures are "adequate" to ensure safety.
Syrian refugees are singled out as "detrimental to the interests of the United States" and banned from entering the country indefinitely.
The US may admit refugees on a case-by-case basis during the freeze, and the government will continue to process requests from people claiming religious persecution, "provided that the religion of the individual is a minority religion in the individual's country."
The order suspended a resettlement programme that allowed 85,000 people fleeing war, hunger, and political or religious persecution, to be resettled in the US last year.
Paul Ryan, the republican speaker of the House, said it was "time to re-evaluate and strengthen the visa-vetting process."
- Originally published in Telegraph UK
