“If this continues for two years obviously it’ll be problematic,” he said, before offering a more hopeful, if still revealing, note: “He has a habit of responding to his impulses but then those are tempered.”

Yet that unpredictable behavior is what has so many Republicans alarmed at the impending departures of Mr. Mattis and the White House chief of staff, John F. Kelly.

For the first half of this administration, traditional Republicans and many mainstream G.O.P. voters had been able to assuage their reservations about Mr. Trump by looking at the economy’s steady growth, the president’s growing bond with congressional Republicans and, perhaps most important of all, the sturdy team of military officers guiding him on national security.

But those totems of normalcy disappeared this week, or seemed to, as Mr. Trump suddenly signaled troop withdrawals from Syria and Afghanistan, Mr. Mattis resigned in protest, the federal government headed toward a shutdown and the stock market fell sharply.

Perhaps no Republicans are more on edge than those who will be on the ballot with the president in 2020. Senator Cory Gardner of Colorado, perhaps the most vulnerable Republican senator up for re-election, suggested the president needed to recognize the damage he was inflicting with the suburban voters who are so pivotal to the party.

“I think conduct matters and I think message matters and we need to do a better job,” Mr. Gardner said.

Lanhee Chen, a Hoover Institution fellow who advised Mitt Romney’s presidential campaign in 2012, said Mr. Trump was sorely testing the strength of his political support. Mr. Chen, who counsels a number of influential congressional Republicans, said Mr. Trump’s drastic shift in the national security arena had the potential to be destabilizing on the right.