The move, which also temporarily bars refugees from entering the US, ushers in the first clash between Trump's populist and proudly politically incorrect leadership style and what his critics see as bedrock values that define the nation.
It also represents a pronounced shift in US counter-terrorism tactics and a sharp turn in the debate underpinning national security policy since the 9/11 attacks -- how to best keep Americans safe and battle extremism at home and abroad without alienating Muslims in a way that could foster more radicalism.
Meanwhile, legal challenges to Trump's moves herald the start of a constitutional showdown between the president's vision of an expansive activist presidency and the checks and balances built into the American political system meant to curb executive power. Legal stays issued by several judges against the measures are likely to be the first actions in a long, complicated legal challenge to the moves that seem almost inevitable to wind up at the Supreme Court.
Trump rejected the storm of criticism whipped up by his immigration measures, blaming long waits at the airport for detained travelers on Delta computer outages -- which were confined to Sunday evening -- as well "the tears of (Sen. Chuck) Schumer
"There is nothing nice about searching for terrorists before they can enter our country. This was a big part of my campaign. Study the world!" he tweeted
The White House earlier made clear that Trump is deeply committed to the new policy, despite the raging controversy. CNN's Jake Tapper reported
that Stephen Miller, a senior policy adviser to Trump, urged officials in the Departments of State, Homeland Security and others to disregard what he described as hysteria on television over the order.
Other Trump supporters argued that the president has a mandate following a campaign in which he made his immigration views clear.
"I believe that President Trump is doing exactly what he campaigned on. He was elected President and now he is delivering the goods. People in America want to be safe," former Arizona Gov. Jan Brewer said on CNN's "State of the Union."
One senior administration official noted that the immigration move is another promise kept by Trump to his supporters following a week in which the president also addressed trade and health care through executive measures.
"It should come as no surprise to anybody that the first week went exactly as he said it would," the official said, speaking on condition of anonymity to discuss the White House's tactics.
The official argued that the temporary suspensions of entry to visitors from Iran, Iraq, Syria, Sudan, Somalia, Libya and Yemen are a reasonable measure to allow time for a new system of vetting to be introduced.
The White House dismisses the idea that the measures amount to a ban on Muslims.
Darker moments in history
But the targeting of nations that are overwhelmingly Muslim recalls some of the darker moments of American history when individuals or groups were singled out for special scrutiny.
Only time will tell if Trump's travel ban comes to be regarded in the same light as the Alien and Sedition Acts signed into law by President John Adams in the late 18th century, which made it easier to deport foreigners and imposed restrictions on new immigrants.
Other historic parallels could include the internment of Japanese Americans during World War II and the witch hunts by Senator Joseph McCarthy against alleged communists in politics and the arts in the 1950s.
The order's indefinite ban on Syrian refugee entries -- on the grounds that it is impossible to vet people from the shattered nation for terrorism links -- also means the United States is in effect turning its back on the victims of the world's worst current humanitarian crisis.
Both Presidents George W. Bush and Barack Obama took great care to avoid the idea that the United States was stigmatizing all Muslims while they pursued ruthless anti-terrorism campaigns against Islamic radicals.
That led critics, especially of Obama, to argue that US policy was handicapped by nuance and political correctness and an unwillingness to admit there was a threat posed by "radical Islamic terrorism."
Trump's actions obliterate that approach.
"'Extreme vetting' is just a euphemism for discriminating against Muslims," said Anthony Romero, the executive director of the American Civil Liberties Union in a statement.
"Identifying specific countries with Muslim majorities and carving out exceptions for minority religions flies in the face of the constitutional principle that bans the government from either favoring or discriminating against particular religions," he said.
Senior Democrats also say Trump's measures conflict with core American values and will be ineffective as a device in cracking down on terrorism.
"This order contravenes the principles of religious liberty, equality, and compassion that our nation was founded upon in its discriminatory impact of Muslims," said Adam Schiff, the senior Democrat on the House Intelligence Committee.
"It also plays into the Al Qaeda and ISIS narrative that the West is no place for Muslims and that we are engaged in a war of civilizations."
As the day wore on Sunday, a number of Republicans added their voices to the criticism of the content of Trump's executive order and the way it was rolled out.
Nebraska GOP Sen. Ben Sasse, a frequent Trump critic, said Trump was right to reject the notion that terrorism had no connection to Islam or certain countries, but argued the travel ban was too broad.
"If we send a signal to the Middle East that the US sees all Muslims as jihadis, the terrorist recruiters win by telling kids that America is banning Muslims and that this is America versus one religion," Sasse said.
Significantly, Sen. Bob Corker, who chairs the Senate Foreign Relations Committee, also weighed in, saying the program had been poorly implemented, especially with regard to green card holders.
"The administration should immediately make appropriate revisions," Corker said. The travel bans were also criticized by Republican senators Lindsey Graham, John McCain and Cory Gardner.
One of the few lawmakers to back Trump's move, GOP Rep. Devin Nunes, who chairs the House intelligence Committee, described it as a "useful" temporary measure to help authorities verify who was coming into the US.
The speed with which Trump instituted the travel ban has fundamentally reshaped America's course in the world and speaks to the profound political change unfolding early in his presidency.
His approach towards Muslim nations appears to be a direct repudiation of Obama's warning in a farewell speech about the new president's instincts.
"Democracy can buckle when it gives into fear. So just as we as citizens must remain vigilant against external aggression, we must guard against a weakening of the values that make us who we are," Obama said in Chicago earlier this month, arguing he had put the war on terror on a firmer legal footing.
Damaging the image of the US
The immigration measures could also damage the image of the United States around the world and expose allied foreign leaders to significant political pressure when they cooperate with Trump.
German Chancellor Angela Merkel told Trump in a telephone call Saturday that the halt to refugee entries from Syria infringed on the Geneva Conventions, her spokesman said.British Prime Minister Theresa May, who was pictured holding hands with Trump at the White House hours before he signed the executive order on Friday, faced a backlash at home.
While Trump's action raises philosophical questions, it is also focuses attention on the new administration's approach to countering terrorism and its basic competence after the orders triggered chaos.
It is, for instance, arguable whether the orders cover the gravest threats to US security. Most of the attacks on US soil since 9/11 have been the work of American citizens or legal permanent residents who had no formal training by groups like ISIS or Al-Qaeda.
For example, the mass killing at a gay nightclub in Orlando last June was carried out by Omar Mateen, who was born in New York to parents who immigrated to the US from Afghanistan. Foreigners who apply for permanent residency already have to undergo a rigorous screening process.
Trump's executive order on immigration warned of the need to learn the lessons of the September 11, 2001, attacks. Yet the 9/11 hijackers were citizens of Saudi Arabia, the UAE, Lebanon and Egypt -- nations not covered by the order.
Other countries where terrorism is a simmering problem, like Pakistan, were also missing.
Many terrorism experts believe the most serious threat posed by foreigners intending to conduct terrorism on US soil could come from fighters who waged war in Syria and have European Union passports and could enter America without a visa. The order does nothing to address that issue.
In practical terms, the sudden change to US policy on immigrants and refugees also provoked chaos and confusion, with airport staff uncertain about what the orders meant, and some travelers detained or refused entry to the US.
It is not clear that relevant departments of government were consulted or informed about the implications of the order or if senior administration appointees had any input into the decision. That scenario casts light on the way the White House is already imposing its power on the rest of the administration.
The result on Saturday was administrative chaos and a sense of uncertainty about what Trump's executive orders mean or how they will be implemented that is becoming a trend for the young administration.
"This was an extreme vetting program that wasn't properly vetted," Ohio Republican Sen. Rob Portman told CNN's "State of the Union."