Still, Mr. Trump’s executive order — which froze all refugee arrivals in the United States and barred the entry of citizens of Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen for 90 days — has sent a whirlwind of confusion, anxiety and fury across the Middle East and Africa. Refugees have been turned back at airports, families separated indefinitely and long-planned trips upended.

“I thought in America, there were institutions and democracy,” said Fuad Sharef, 51, an Iraqi Kurd bound for New York who was turned away from the Cairo airport with his wife and three children on Saturday morning. “This looks like a decision from a dictator. It’s like Saddam Hussein.”

On Sunday, Trump administration officials backtracked on one aspect of the order, saying green-card holders would be allowed to return to the United States. In a Facebook post on Sunday evening, Mr. Trump insisted that his policy was not a “Muslim ban” and accused the news media of inaccurate reporting. Hours earlier, he had characterized the conflict with the Islamic State in starkly sectarian terms, asserting on Twitter: “Christians in the Middle East have been executed in large numbers. We cannot allow this horror to continue!”

In fact, a majority of the Islamic State’s victims have been Muslims, many of them shot, burned or beheaded. Among the Muslims who managed to escape Islamic State territory are the refugees Mr. Trump has now excluded.

In a phone conversation with Mr. Trump on Saturday, Chancellor Angela Merkel of Germany cited the 1951 Refugee Convention, which calls on signatories to take in people fleeing war, according to Steffen Seibert, Ms. Merkel’s spokesman. Yet in much of the Middle East, Mr. Trump is less likely to get such a scolding.

He has drawn close to Mr. Sisi of Egypt, whom he called a “fantastic guy,” and is considering designating the Muslim Brotherhood, Mr. Sisi’s sworn enemy, a terrorist organization. In a call last week, the two leaders discussed a possible visit to the White House by Mr. Sisi, whose administration faces accusations of human rights abuses — an unthinkable prospect during the Obama administration.

In his order on Friday, whose stated aim is to keep extremists out of the United States, Mr. Trump invoked the Sept. 11 attacks three times. Yet Saudi Arabia, which was home to 15 of the 19 attackers, was not included on the list of countries whose citizens would be shut out. That reflects the deep economic and security ties between the United States and Saudi Arabia. Mr. Trump also has a personal financial link: In August 2015, just as his campaign was gathering steam, the Trump Organization registered eight companies in Saudi Arabia that were linked to a hotel development in the city of Jidda.