A visibly disgusted President Trump said Wednesday that he won’t work with Democrats while they are seeking to impeach him, cutting short a meeting with Democratic leaders at the White House and vowing to go it alone until Democrats finish their “phony investigations.”

“Let them play their games,” Mr. Trump told reporters in a hastily arranged press conference in the White House Rose Garden after he sent the Democrats packing. “We’re going to go down one track at a time.”

The confrontation threw into doubt Mr. Trump’s trade deal with Canada and Mexico, a potential $2 trillion plan for infrastructure spending, a possible two-year budget deal and budding signs of cooperation from the White House with some congressional investigations.

The prospect of Washington gridlock through the presidential election in November 2020 erupted after House Speaker Nancy Pelosi, California Democrat, accused Mr. Trump of being “engaged in a cover-up” by stonewalling multiple investigations by her troops. She made the comment after emerging from a morning meeting with the Democratic chairmen of five committees investigating the president in the wake of special counsel Robert Mueller’s report into Russian meddling in the 2016 election.

An hour later, Mrs. Pelosi visited the White House with Senate Minority Leader Charles E. Schumer, New York Democrat, for a scheduled meeting on infrastructure and found the president spoiling for a fight.

Aides said the president balked at shaking hands with the Democrats and instead confronted Mrs. Pelosi about her “cover-up” comment. The frosty meeting lasted a few minutes.

“To watch what happened in the White House would make your jaw drop,” Mr. Schumer said. “It was planned. When we got in the room, the curtains were closed. There was a place for [the president] at the front so he could stand and attempt to tell us why he wouldn’t do infrastructure.”

As soon as the president left the Democrats, he went to the Rose Garden, where the White House had summoned the media and set up a podium, platform and signs decrying the Mueller investigation as a partisan waste of money.

“I don’t do cover-ups,” Mr. Trump told reporters. “I told Sen. Schumer, Speaker Pelosi, ‘I want to do infrastructure. I want to do it more than you want to do it. But you can’t do it under these circumstances. So get these phony investigations over with.’”

Mrs. Pelosi and Mr. Schumer said they had gone to the White House despite their antipathy toward the president to work on an infrastructure deal.

“We went in the spirit of bipartisanship to find common ground with the president on this,” Mrs. Pelosi said after her third face-to-face clash with Mr. Trump since December. “Maybe it was lack of competence on his part that he really couldn’t come match the greatness of the challenge that we have.”

Mr. Schumer held up what he said was a 35-page infrastructure plan he was prepared to give to Mr. Trump, but he told reporters that the president wanted to blow up the process instead.

Major business and labor groups have been pushing for a package from Washington to rebuild roads, bridges, ports, rural broadband networks and other infrastructure. U.S. Chamber of Commerce President Thomas J. Donohue called the confrontation a distraction.

“Every day, the cost of federal inaction grows,” Mr. Donohue said. “We urge Congress and the president to immediately resume discussions on America’s failing infrastructure. The time to act is now.”

Mr. Schumer said he doubted the president’s explanation that Mrs. Pelosi’s comments about a cover-up sparked the president’s fury. He said Mr. Trump instead appeared to be looking for reasons to refuse to work together.

“There were [House] investigations going on three weeks ago when we met,” Mr. Schumer said. “And he still met with us. But now that he was forced to actually say how he’d pay for [infrastructure], he had to run away. It’s clear that this was not a spontaneous move on the president’s part.”

Mrs. Pelosi said the president “just took a pass” on working with them.

“I pray for the president of the United States,” she told reporters.

At their meeting about infrastructure three weeks ago, the president didn’t raise the issue of multiple Democratic investigations. On Wednesday, however, he recited his lengthy and unprecedented cooperation with the special counsel’s investigation, which found no collusion between Russia and the Trump campaign in 2016.

Mr. Mueller didn’t reach a conclusion on obstruction of justice, but Attorney General William Barr determined from the report that the president didn’t engage in obstruction.

“I’m the most transparent president probably in the history of the country,” Mr. Trump said.

“I let the White House counsel speak for 30 hours. Thirty hours. Five hundred witnesses that I allowed to testify. And then I have Nancy Pelosi go out and say that the president of the United States engaged in a cover-up. We did nothing wrong.”

Mr. Trump’s statement threatened to derail a two-year spending deal between the White House and bipartisan leaders in Congress that was nearing completion. Before the president’s press conference, congressional leaders were optimistic that they could reach a deal to raise spending caps by the end of the week.

It also could upend his replacement for the North American Free Trade Agreement and talks on infrastructure spending. The president told Democratic leaders Tuesday night that he wanted Congress to approve his trade deal with Canada and Mexico before he would agree to a $2 trillion infrastructure package.

Rep. John A. Yarmuth, Kentucky Democrat, called Mr. Trump’s statement “essentially blackmail” and said it could poison the conversations on spending and debt, too.

Still, Mr. Yarmuth, chairman of the House Budget Committee, said Mr. Trump has a habit of switching stances quickly so it’s not clear whether there is lasting damage.

“It wouldn’t shock me if it was back at the table at 8 o’clock tonight saying let’s make a deal,” he said.

During a visit to Charlotte, North Carolina, host of the 2020 Republican National Convention, Vice President Mike Pence said congressional Democratic leaders “are more interested in playing politics than making real progress for the American people.”

“An hour before Democrats were scheduled to go to the White House to talk about rebuilding our nation’s infrastructure, the speaker of the House accused the president of the United States of ‘engaging in a cover-up,’” Mr. Pence told the audience at a mill. “Enough is enough.”

The vice president said “reckless accusations won’t pave a single new road. Partisan investigations won’t create a single new job.”

“It’s time for Congress to end these partisan games and work with us for the good of every American,” he said.

Others say Congress can, and often does, work that way.

“Congress can indeed walk and chew gum at the same time,” said Ross Baker, a political science professor at Rutgers University in New Jersey. “The bulk of the investigative activity is restricted to three committees: intelligence, oversight and Judiciary. Transportation and Infrastructure is doing roads and bridges, and Energy and Commerce is marking up prescription drug legislation. Ways and Means isn’t really doing much in hearings but is pressing the White House for the Trump tax returns.”

Some Democrats criticized the president’s reaction to the “cover-up” accusation, noting that former Trump attorney Michael Cohen is serving time in federal prison after pleading guilty to eight federal crimes, including campaign finance violations, related to payments to porn actress Stormy Daniels to cover up an affair she says she had with Mr. Trump. Federal prosecutors said Cohen made those payments “in coordination with and at the direction of” Mr. Trump, an accusation that the president has consistently denied.

Mr. Trump’s frustration with endless investigations and with former Obama administration officials who authorized spying on his campaign was evident — perhaps more so than at any other time of his presidency. He said at one point that his son, Donald Jr., is “a good young man who’s gone through hell” from the investigations.

“We’ve been doing this since I’ve been president,” Mr. Trump said. “And actually, the crime was committed on the other side. We’ll see how that all turns out. This whole thing was a takedown attempt at the president of the United States.”

He refused to say the word “impeachment” and instead referred to Democrats’ pursuit of “the big ‘I’ word.”

“I’ll tell you, there’s a danger here,” Mr. Trump said. “If someday a Democrat becomes president and you have a Republican House, they could impeach him for any reason, or her. Any reason. We can’t allow that to happen.”

The White House press conference was unusual and put Mr. Trump’s showmanship on full display.

White House officials told the entire press corps to gather for an impromptu event, before leading reporters into the Rose Garden.

A sign on the president’s lectern rattled off statistics about the Mueller investigation, including “2,800 subpoenas,” “500+ witnesses,” “675 days” and “18 angry Democrats.”

Mr. Trump held up a piece of paper to underscore those numbers, and White House staff later handed out copies to reporters.

Key White House aides — including acting Chief of Staff Mick Mulvaney, counselor Kellyanne Conway and press secretary Sarah Huckabee Sanders — watched the spectacle from the wings.

⦁ Tom Howell Jr. and Gabriella Muñoz contributed to this report.

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