The order also effectively bans all people with passports from Iran, Iraq, Somalia, Libya, Sudan and Yemen from entering the country for three months, and it bans Syrians indefinitely.
A Financial Times editorial this morning argues that chief executives must speak up, having been "compliant" in Trump's agenda to date.
Some have, with high profile examples detailed below:
Mark Zuckerberg - Facebook
Facebook CEO Mark Zuckerberg, who expressed concern about the impact of the executive orders signed by Trump, said the US should be proud that it is a nation of immigrants.
"We are a nation of immigrants, and we all benefit when the best and brightest from around the world can live, work and contribute here. I hope we find the courage and compassion to bring people together and make this world a better place for everyone," he said.
Brian Chesky - AirBnB
Brian Chesky, CEO and co-founder of short-term property rental business AirBnB, said on Twitter "we must stand with those affected". The company is offering free housing to refugees and anyone who is not allowed into the US. It has taken a similar measure on previous occasions, including after the Paris terror attacks of late 2015.
Howard Schultz - Starbucks
In a letter to staff on Sunday (29 January), Howard Schultz, chairman and CEO of coffee retailer Starbucks, said the company would hire 10,000 refugees over the next five years in response to Trump's latest order.
He said the company would hire people across the 75 countries where Starbucks does business, but would start this effort in the US.
Elon Musk - SpaceX/Tesla Motors
SpaceX founder and Tesla Motors engineer Elon Musk has said that many people negatively affected by the travel ban do not deserve to be rejected.
He has therefore asked his followers on Twitter to read the immigration order and suggest specific amendments, which he will then present to the President.
Musk, alongside Uber CEO Travis Kalanick and several other business leaders, is part of the Trump's economic advisory group.
Please read immigration order. Lmk specific amendments. Will seek advisory council consensus & present to President. https://t.co/qLpbsP4lEk— Elon Musk (@elonmusk) January 29, 2017
Travis Kalanick - Uber
Ride sharing app Uber came under fire last week after social media users accused it of being too lenient with Trump. #BoycottUber trended on Twitter after its CEO Travis Kalanick said the company was willing to work with the Trump administration.
A day later, Kalanick wrote to staff opposing the travel ban and offered to compensate drivers affected by it for the next three months.
Our CEO's reaction to immigration order: "We'll compensate drivers impacted by the ban pro bono for next 3 months." https://t.co/meCT1ahEjH— Uber (@Uber) January 29, 2017
The mayor of London took to Facebook (and then Twitter), dubbing the ban "shameful and cruel".
"While every country has the right to set its own immigration policies, this new policy flies in the face of the values of freedom and tolerance that the USA was built upon," he said yesterday.
He added that he would work with the Government on behalf of Londoners affected.
Muslim Council of Britain
The MCB has condemned the executive order and has called on the UK Government to "speak out much more forcefully and stand up for british values it supposedly seeks from others".
Harun Khan, secretary general of the MCB, said: "The ban will affect us here in Britain, as those with dual nationality such as Sir Mo Farah and Conservative MP Nadhim Zahawi will also be affected by this ban. Our government should express in no uncertain terms how daft this policy is to its US counterparts, and press home how counter-productive it is in its professed fight to confront terrorism."
Since taking office on 20 January, Trump has signed several executive orders, which give the US President power to create laws with the same legal influence as those passed by the US Congress.