The US is facing legal action and calls to reverse his sweeping immigration restrictions as the measures come into effect at Irish airports and refugees have been detained in the US.
However, President Trump has said his executive order curbing immigration into the United States was not a ban on Muslims and was working out well.
"It's not a Muslim ban," Trump told reporters in the Oval Office at the White House. "It's working out very nicely. You see it at the airports, you see it all over."
The new Republican president on Friday put a four-month hold on allowing refugees into the United States and temporarily barred travelers from Syria and six other Muslim-majority countries. He said the moves would protect Americans from terrorism, in a swift and stern delivery on a campaign promise.
"We're going to have a very, very strict ban and we're going to have extreme vetting, which we should have had in this country for many years," Trump said.
On Saturday he also signed an executive action calling for a military plan to be drafted in the next 30 days, outlining how to defeat Islamic State.
"It is going to be very successful," Trump said as he signed the order in the Oval Office at the White House. A copy of the order was not immediately available but was expected to be released later.
In his first week in office Trump put a four-month hold on allowing refugees into the United States and temporarily barred travelers from Syria and six other Muslim-majority countries, saying the moves would help protect Americans from terrorist attacks.
He said his most sweeping use of his presidential powers would give his administration time to develop more stringent screening procedures for refugees, immigrants and visitors.
"I'm establishing new vetting measures to keep radical Islamic terrorists out of the United States of America. Don't want them here," Trump said earlier on Friday at the Pentagon.
"We only want to admit those into our country who will support our country and love deeply our people," he said.
The US Embassy in Dublin has advised people from the proscribed list of nations that the restrictions are in force at Irish airports. Officials have also advised those who had booked visa interviews not to attend.
"The Executive Order suspends visa issuance and entry into the United States of nationals of countries of particular concern (Iraq, Iran, Libya, Somalia, Sudan, Syria, and Yemen), including dual nationals of these countries," the embassy said in a statement.
"US Customs and Border Protection has implemented this provision at its Pre-Clearance facilities, including at Dublin and Shannon airports, as well as at Ports of Entry across the United States. Nationals of these countries, including dual nationals, who have already scheduled a visa interview at U.S. Embassy Dublin should not attend their appointment as we will not be able to proceed with the visa interview.
"We will announce any additional changes affecting travelers to the United States as soon as that information is available."
However, the embassy would not be drawn on whether any people had been denied access to US-bound flights.
Meanwhile, the UN and the International Organization for Migration (IOM) has called on President Donald Trump to continue the US' resettlement program which is "one of the most important in the world" for refugees.
The two agencies have issued a joint statement urging the US to continue allowing refugees entry.
"The needs of refugees and migrants worldwide have never been greater, and the U.S. resettlement program is one of the most important in the world," the agencies said.
"The longstanding U.S. policy of welcoming refugees has created a win-win situation: it has saved the lives of some of the most vulnerable people in the world who have in turn enriched and strengthened their new societies. The contribution of refugees and migrants to their new homes worldwide has been overwhelmingly positive."
In the US immigration lawyers have filed a lawsuit to block President Donald Trump's order halting the entry of refugees and foreign nationals from seven predominantly Muslim countries to the United States, saying numerous people have already been unlawfully detained.
The lawyers from numerous immigration organizations and the American Civil Liberties Union sued in federal court in Brooklyn, New York, on behalf of two Iraqi men, one a former U.S. government worker and the other the husband of a former U.S. security contractor.
The two men had visas to enter the United States but were detained on Friday night at New York's John F. Kennedy International Airport, hours after Trump's executive order put a four-month hold on allowing refugees into the United States and temporarily barred travelers from Syria and six other Muslim-majority countries, the lawsuit said.
One of the plaintiffs, Hameed Khalid Darweesh, 53, worked for the U.S. Army and for a U.S. contractor in Iraq from 2003 to 2013 as an interpreter and engineer, the lawsuit said.
The other plaintiff, Haider Sameer Abdulkhaleq Alshawi, 33, is the husband of a woman who worked for a U.S. security contractor from 2006 to 2007 as an accountant, the lawsuit said.
Their "continued unlawful detention is part of a widespread pattern applied to many refugees and arriving aliens detained after the issuance of the January 27, 2017 executive order," the lawyers wrote.
Shortly before 18.00 on Irish time on Saturday Reuters reported that one of the two men, Hameed Khalid Darweesh, had been admitted entry into the US.
Meanwhile, five Iraqi passengers and one Yemeni were barred from boarding an EgyptAir flight from Cairo to New York on Saturday sources at Cairo airport said.
The passengers, arriving in transit to Cairo airport, were stopped and re-directed to flights headed for their home countries despite holding valid visas, the sources said.
The bans, though temporary, took effect immediately, causing havoc and confusion for would-be travelers with passports from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
Dutch airline KLM also said on Saturday it had refused carriage to the United States to seven passengers from the countries who are included in the ban.
A spokeswoman for KLM, part of the Franco-Dutch Air France KLM group, declined to specify which countries the passengers came from or where they were flying from.
"Worldwide, we had seven passengers whom we had to inform that there was no point in us taking them to the US.," said spokeswoman Manel Vrijenhoek. "There is still some lack of clarity about whom this ban affects.
A spokesperson for Homeland security said that even those who are legal US citizens were to be subjected to the new rules.
People holding so-called green cards, making them legal permanent U.S. residents, are included in President Donald Trump's executive action temporarily barring people from seven Muslim-majority countries from entering the United States, a Department of Homeland security spokeswoman said on Saturday.
"It will bar green card holders," Gillian Christensen, acting Department of Homeland Security spokeswoman, said in an email.
However, the UN and the IOM have urged the US to continue leading the way in providing resettlement for vulnerable refugees and migrants.
"Resettlement places provided by every country are vital. UNHCR, the UN Refugee Agency, and IOM, the International Organization for Migration, hope that the U.S. will continue its strong leadership role and long tradition of protecting those who are fleeing conflict and persecution," their statement reads.
"UNHCR and IOM remain committed to working with the U.S. administration towards the goal we share to ensure safe and secure resettlement and immigration programmes.
"We strongly believe that refugees should receive equal treatment for protection and assistance, and opportunities for resettlement, regardless of their religion, nationality or race.
"We will continue to engage actively and constructively with the U.S. Government, as we have done for decades, to protect those who need it most, and to offer our support on asylum and migration matters.
France and Germany have also expressed concern over the restrictions on immigrations.
"Welcoming refugees who flee war and oppression is part of our duty," French Foreign Minister Jean-Marc Ayrault said at a joint news conference with German counterpart Sigmar Gabriel.
"The United States is a country where Christian traditions have an important meaning. Loving your neighbor is a major Christian value, and that includes helping people," said Gabriel.
"I think that is what unites us in the West, and I think that is what we want to make clear to the Americans."
Iran has condemned visa ban against Tehran and six other majority-Muslim countries as an "open affront against the Muslim world and the Iranian nation" and vowed to retaliate.
A Foreign Ministry statement carried by state media said Iran "would take appropriate consular, legal and political measures" against the ban, which was announced by President Donald Trump on Friday.
Additional reporting by Reuters