WASHINGTON — Travelers were stranded around the world, protests escalated in the United States and anxiety rose within President Trump’s party on Sunday as his order closing the nation to refugees and people from certain predominantly Muslim countries provoked a crisis just days into his administration.

The White House pulled back on part of Mr. Trump’s temporary ban on visitors from seven countries by saying that it would not apply to those with green cards granting them permanent residence in the United States. By the end of the day, the Department of Homeland Security formally issued an order declaring such legal residents exempt from the order.

But the recalibration did little to reassure critics at home or abroad who saw the president’s order as a retreat from traditional American values. European leaders denounced the order, and some Republican lawmakers called on Mr. Trump to back down. As of Sunday evening, officials said no one was being held at American airports, although lawyers said they believed that dozens were still being detained.

More than any of the myriad moves Mr. Trump has made in his frenetic opening days in office, the immigration order has quickly come to define his emerging presidency as one driven by a desire for decisive action even at the expense of deliberate process or coalition building. It has thrust the nine-day-old administration into its first constitutional conflict, as multiple courts have intervened to block aspects of the order, and into its broadest diplomatic incident, with overseas allies objecting.