The White House and Democratic leaders on Capitol Hill fumed Thursday over Senate Republicans yanking their support from a proposed congressional commission to investigate the Jan. 6 riot at the U.S. Capitol, as reality set in that the plan faced all but certain defeat.

White House press secretary Jen Psaki and House Speaker Nancy Pelosi alluded to a Democrat-led investigation if Republicans refused a bipartisan investigation on the storming of the Capitol by a pro-Trump mob.

Republicans, including Senate Majority Leader Mitch McConnell, balked at creating the commission because they said it would be a Democratic political hit job aimed at former President Donald Trump and the GOP.

Ms. Psaki expressed dismay that so many Republicans are not “doing what’s right.”

“The attack on the Capitol on Jan. 6 was an unprecedented assault on our democracy and demands a full, independent investigation into what happened,” she said. “This is not a political issue, in the president’s view. This is a question of how we secure our democracy and the rule of law.”

“They have the opportunity to do that. If they don’t … we’ll have a conversation about it,” Ms. Psaki said.

Earlier this year, the White House said Mr. Biden supported an investigation but would leave it up to Congress to decide.

Mrs. Pelosi has suggested a backup plan to create a Democrat-led investigatory panel if Senate Republicans block the proposal for a 9/11-style commission to examine what happened on Jan. 6.

“If they don’t want to do this, we will find the truth,” she said Wednesday after the House passed the commission bill and sent it toward certain peril in the Senate.

Senate Majority Leader Charles E. Schumer on Thursday railed against Republicans for moving “the goalposts” for a bipartisan agreement to form an independent commission.

“In the aftermath of the Capitol attack, the Republican leader, Senator [Mitch] McConnell, said we needed a serious and thorough review of the attack,” the New York Democrat said. “But very quickly, the goalposts started to move. … Republicans started complaining that Democratic proposals for establishing a commission were too partisan.”

Democrats insist they agreed to all Republican demands, including having an equal number of Democratic and Republican members and giving both sides subpoena power.

The proposed panel would consist of five Democratic and five Republican appointees and would face a 2021 deadline to complete its work.

“The real reason, it seems, Republican leaders are suddenly opposed to this bipartisan commission is they don’t want to talk about the ‘big lie’ at all,” said the majority leader. “They don’t even want to investigate how former President Donald Trump instigated an attack on our democracy because he was angry about losing the 2020 election.”

Mr. McConnell, Kentucky Republican, announced his opposition shortly after it passed the House — a move that gave a green light for Senate Republicans to kill the commission.

“After careful consideration, I’ve made the decision to oppose the House Democrats’ slanted and unbalanced proposal for another commission to study the events of January the 6th,” Mr. McConnell said.

The House approved the creation of a commission in a 252-175 vote, with 35 Republicans joining all the chamber’s Democrats in support.

Mr. Trump bemoaned that 35 Republicans broke ranks.

“See, 35 wayward Republicans — they just can’t help themselves,” he said in a statement. “We have much better policy and are much better for the Country, but the Democrats stick together, the Republicans don’t.”

Mr. Trump said Democrats “don’t have the Romney’s, Little Ben Sasse’s, and Cheney’s of the world. Unfortunately, we do.” It was a reference to Sen. Mitt Romney of Utah, Sen. Ben Sasse of Nebraska and Rep. Liz Cheney of Wyoming, all Republicans who have clashed with Mr. Trump.

“Sometimes there are consequences to being ineffective and weak. The voters understand!” the former president said.

The attack on the Capitol, in which the pro-Trump mob temporarily delayed confirmation of Mr. Biden’s election win, has become a defining moment for the two parties. Democrats refer to it as an insurrection against democracy. In response, they have made a top priority of searching out and combatting the threat they see from White supremacists and domestic terrorists.

Republicans, though denouncing the attack, view it as a protest that got out of control. They accuse Democrats of seeking political gain from the attack, which directly resulted in the deaths of four people, all of them Trump supporters.

Rep. Jim Jordan summed up Republicans’ objections to the commission.

The Ohio lawmaker, the top Republican on the Judiciary Committee, labeled it “Impeachment 3.0.”

“This is one more chance for them to go after President Trump because … they don’t want President Trump to run in 2024,” he told Fox News host Maria Bartiromo on “Mornings with Maria.”

He contended that the commission was designed by Democrats to smear Mr. Trump and distract from the mounting troubles surrounding the Biden administration.

“We went from energy independence to gas lines. We went from a secure border to chaos. We went from peace in the Middle East to rockets being fired on our ally Israel. That’s what 120 days of the Biden administration will give you,” he said. “They want to talk about anything else, so why not go after President Trump again? That’s what this is all about.”