Three months after the federal appeals court in California blocked President Trump’s first travel ban, a three-judge panel of the same court heard arguments on Monday in a challenge to Mr. Trump’s revised ban, this one limiting travel from six predominantly Muslim countries.

Mr. Trump views the appeals court — the United States Court of Appeals for the Ninth Circuit — as a hostile forum, frequently criticizing it on Twitter. But the judges hearing Monday’s arguments asked probing questions of both sides and may not be ready to endorse the sweeping reasoning of a federal judge in Hawaii who blocked major parts of the revised order.

The appeals court judges, all appointed by President Bill Clinton, barely tipped their hands, but they did say they recognized that the dispute before them was momentous. The argument was held in Seattle and broadcast live on C-Span.

“This court is very well aware of the importance of this case,” Judge Ronald M. Gould said at the outset.

In making the administration’s case, Jeffrey B. Wall, the acting United States solicitor general, urged the court to ignore campaign statements from Mr. Trump concerning his intention to issue a “Muslim ban” and focus instead on the terms of the revised executive order.

“The order on its face doesn’t have anything to do with religion,” he said.

Judge Michael Daly Hawkins asked whether Mr. Trump had ever disavowed his campaign statements.

Mr. Wall responded that “over time the president clarified that what he was talking about were Islamic terrorist groups and the countries that shelter or sponsor them.”

But Neal K. Katyal, a lawyer for Hawaii, which is challenging the revised order, said that Mr. Trump’s campaign website until very recently continued to feature a 2015 news release that said “Donald J. Trump is calling for a total and complete shutdown of Muslims entering the United States.”

That document, Mr. Katyal said, “just happened to disappear moments before the Fourth Circuit argument last week.”

On May 8, a 13-judge panel of the United States Court of Appeals for the Fourth Circuit, in Richmond, Va., heard arguments in a separate appeal of an order blocking part of the revised travel ban, that one from a federal judge in Maryland.

Judge Richard A. Paez asked Mr. Katyal whether Mr. Trump “is forever barred from issuing an executive order along these lines.”

Mr. Katyal responded that a revised order might be proper if Congress enacted new legislation authorizing it, if Mr. Trump explicitly disavowed his earlier statements or if he described Islam in the positive terms that President George W. Bush used after the Sept. 11 attacks.

Mr. Wall argued that the Constitution and the immigration laws give the president vast authority to make national security judgments about who may enter the country. The role of the courts, he said, was limited to determining whether the government had a “facially legitimate and bona fide reason” for taking action.

Late in the argument, Judge Hawkins asked Mr. Katyal, “Why shouldn’t we be deferential to the office of president of the United States on these issues?”

“That’s the million-dollar question,” Mr. Katyal said — and he added that the First Amendment’s ban on government establishment of religion supplied the answer.

“Is this executive order, viewed from the standpoint of an objective observer, an establishment of a disfavored religion, Islam?” Mr. Katyal asked. The answer, he said, was yes.

Mr. Trump issued his initial order on Jan. 27, a week into his presidency. Less than two weeks later, a different three-judge panel of the Ninth Circuit affirmed an order halting it.

Within minutes of the ruling, Mr. Trump vowed to fight it. “SEE YOU IN COURT, THE SECURITY OF OUR NATION IS AT STAKE!” Mr. Trump wrote on Twitter.

But the president did not appeal to the Supreme Court. Instead, he issued a revised executive order.

The new order’s 90-day suspension of entry from Iran, Libya, Somalia, Sudan, Syria and Yemen was more limited and subject to case-by-case exceptions. It omitted Iraq, which had been listed in the earlier order, and it removed a complete ban on Syrian refugees. And it deleted explicit references to religion. Like the earlier order, the new one suspended the nation’s refugee program for 120 days and reduced the annual number of refugees to 50,000 from 120,000.

Hawaii sued to challenge the revised executive order, saying it was the product of religious hostility. On March 15, Judge Derrick K. Watson, of the Federal District Court in Hawaii, blocked major parts of the revised order, saying they violated the Constitution’s ban on government establishment of religion. Judge Watson wrote that the statements of Mr. Trump and his advisers made clear that his executive order amounted to an attempt to treat Muslims with disfavor.

“A reasonable, objective observer — enlightened by the specific historical context, contemporaneous public statements and specific sequence of events leading to its issuance — would conclude that the executive order was issued with a purpose to disfavor a particular religion,” Judge Watson wrote.

Much of Monday’s argument concerned how to reconcile two parts of the immigration laws.

One provision allows the president to “suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants” if he determines that their entry “would be detrimental to the interests of the United States.”

But another provision appears to limit that power, saying that “no person shall receive any preference or priority or be discriminated against in the issuance of an immigrant visa because of the person’s race, sex, nationality, place of birth or place of residence.”

Mr. Wall said the second provision would at most require the issuance of visas to people who could nonetheless still be barred at the border.

Judge Hawkins said that “would be like Tom Hanks at the airport,” apparently referring to “The Terminal,” a 2004 film about a man stranded at Kennedy Airport.