President Trump’s surprise proposal to seize control of Gaza upended decades of bipartisan foreign policy consensus over the United States’ role in the region and tested the sprawling coalition that sent Mr. Trump to the White House last fall.

The president’s words — delivered at a 40-minute news conference on Tuesday night with classic Trumpian bombast — exploded the bounds of the country’s debate over one of the world’s most intractable conflicts and immediately set off infighting in both parties.

Already, the Hamas-led Oct. 7 attacks and Israel’s devastating response in Gaza had propelled the war into the center of the 2024 election, fracturing the Democratic Party and creating improbable new political alliances as some Arab American and Jewish voters shifted to the right.

But a new wave of recriminations arrived on Wednesday, even as skepticism abounded about the president’s far-fetched proposal and as top Trump administration officials walked back elements of the idea. Politicians and activists traded attacks and fought over whether this vision squared with Mr. Trump’s “America First” brand, which has always revolved around his promise to extricate the nation from foreign wars.

From the right, Mr. Trump received perhaps the sharpest Republican pushback of his nascent presidency, which has so far been defined by his party’s complete fealty to him. Some anti-interventionist lawmakers like Senator Rand Paul of Kentucky bristled at the idea. More hawkish leaders, like Senator Lindsey Graham of South Carolina, suggested that the proposal might simply be a nonstarter.

And on the left, some Democrats criticized the Arab and Muslim Americans, progressives and others who opposed Vice President Kamala Harris in protest of the Biden administration’s support for Israel — “is this what you wanted?” the argument went — as others vigorously defended that choice.