Mr Trump tweeted the US needed “extreme vetting, NOW”. His chief of staff said only 109 people, out of 325,000 travelling, had been detained.
A number of judges ruled on the issue – one federal judge temporarily halted the deportation of visa holders.
There has been condemnation from countries around the world.
Mr Trump’s executive order, signed on Friday, halted the entire US refugee program for 120 days, indefinitely banned Syrian refugees, and suspended all nationals from seven Muslim-majority countries for 90 days.
Those who were already mid-flight were detained on arrival – even if they held valid US visas or other immigration permits.
Thousands gathered at airports around the country to protest on Saturday, including lawyers who offered their services for free to those affected.
Further demonstrations are expected on Sunday – including one outside the White House.
Who is affected by the ban?
All travelers who have nationality or dual nationality of Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen are not permitted to enter the US for 90 days, or be issued an immigrant or non-immigrant visa.
This includes those who share dual nationality with allied countries, including the UK, although Canada has been told its dual nationals are not affected.
White House Chief of Staff Reince Priebus said US green-card holders would also not be affected, although he admitted to NBC’s Meet the Press program that they could be subject to greater questioning at airports.
Regarding those held at airports, he said: “Most of those people were moved out. We’ve got a couple of dozen more that remain and I would suspect that as long as they’re not awful people that they will move through before another half a day today.”
He said the seven countries had been chosen because they had already been identified by Congress and the Obama administration as the most likely to harbor terrorists, and did not rule out the fact that more countries could be added to the list.
Some leading Republicans expressed concern.
John McCain called it a “very confusing process” which would “probably, in some areas, give ISIS [Islamic State group] some more propaganda”, while Senate Majority Leader Mitch McConnell said it was important to remember that “some of our best sources in the war against radical Islamic terrorism are Muslims”.
Democrats Senate Majority leader Chuck Schumer said the US now appeared “less humanitarian, less safe, less American” and said the Democrats would introduce legislation to overturn it.
Judges have their say
Late on Saturday, federal Judge Ann Donnelly, in New York, ruled against the removal from the US of people with approved refugee applications, valid visas, and “other individuals… legally authorized to enter the United States”.
The emergency ruling also said there was a risk of “substantial and irreparable injury” to those affected.
She was ruling on a case brought by the American Civil Liberties Union (ACLU) on behalf of two Iraqi men with links to the US military who were detained at JFK Airport in New York.
Both have now been released. Another court hearing is set for February.
Elsewhere in the US:
- In Boston, a judge decided two Iranian nationals, university professors should be released from detention at Logan International Airport.
- An order issued in Virginia banned, for seven days, the deportation of green card holders held at Dulles Airport and ordered the authorities to allow access to lawyers.
- A Seattle judge issued an emergency stay of removal from the US for two people
But the Department of Homeland Security said it would continue to enforce the measures.