House Speaker Nancy Pelosi on Tuesday accused President Trump of a “betrayal of his oath of office” and said Democrats will rise to the challenge of the moment by opening an unprecedented inquiry that could lead to his impeachment.

After months of resisting, Mrs. Pelosi emerged from a meeting with fellow House Democrats to say she was swayed by the latest allegations, reported in the press, that Mr. Trump tried to pressure Ukraine to dig up dirt on Democratic presidential front-runner Joseph R. Biden.

The speaker brushed aside Mr. Trump’s promise to release a transcript Wednesday that he said would prove his innocence. She said the president’s admissions already demonstrate that he has broken the law.

“The times have found us today,” she said. “Not to place ourselves in the same category of greatness as our founders, but to place us in the urgency of protecting and defending our Constitution from all enemies, foreign and domestic.”

No president in modern history has been the target of impeachment in his first term heading into a reelection campaign. President Bill Clinton was impeached in his second term but not removed from office by the Senate. Richard Nixon resigned in his second term while facing certain impeachment.

Mr. Trump, attending the U.N. General Assembly in New York, dismissed the development as “a continuation of the witch hunt.”

Congressional Republicans called Mrs. Pelosi’s move an empty gesture because Democratic committees already said they were acting under the rubric of impeachment. They said Mrs. Pelosi’s announcement didn’t change anything.

“What she said today is no different than what’s been going on,” said Minority Leader Kevin McCarthy, California Republican.

If Mrs. Pelosi’s move didn’t break any legal ground, it did signal a major shift among Democrats, with dozens of members rushing to climb aboard the impeachment train this week in the wake of reporting that Mr. Trump tried to rope Ukraine into boosting his political fortunes.

Hours before Mrs. Pelosi’s announcement, the president authorized the release of an unredacted transcript of his July 25 phone call with Ukrainian President Volodymyr Zelensky. He said the transcript will disprove Democrats’ suspicions that he threatened to withhold $391 million in military aid from Ukraine unless its president agreed to investigate the role of Mr. Biden’s son Hunter in a Ukrainian natural gas company.

“How can you do [impeachment] and you haven’t even seen the phone call?” Mr. Trump said.

The president predicted that the move will backfire on Democrats and be “a positive for me.”

“Our country’s doing the best it’s ever done. They’re going to lose the election,” Mr. Trump told reporters.

Top Democrats said the phone transcript alone won’t be enough to spare Mr. Trump. They also are demanding that the administration turn over to Congress a whistleblower’s complaint against him pertaining to his reported attempt to shake down the Ukrainian president.

“Simply to release the transcript is not going to come close to ending the need of the American public and the Congress to see what actually happened,” said Senate Minority Leader Charles E. Schumer, New York Democrat.

He led the Senate to approve a resolution Tuesday demanding that the complaint be turned over to Congress. Republicans joined in that call, even though allies of the president pointed out that the unidentified whistleblower, a national security employee, wasn’t listening in on the phone call and apparently got the information secondhand.

The White House was preparing to release the whistleblower’s complaint and the inspector general’s report about the complaint, Politico reported Tuesday evening.

Rep. Mark Meadows, North Carolina Republican, tweeted that “amazingly Democrats now say the whistleblower complaint is more important” than the transcript of the phone call.

“Folks, the ‘whistleblower’ wasn’t on the call,” Mr. Meadows tweeted. “They think a secondhand account of the call will tell you more than the *actual call*.”

Mr. Trump acknowledged that he did discuss the Bidens during the call but said the transcript will prove he did not ask Ukraine’s leader for any political favors. He said he withheld the military aid from mid-July until Sept. 11 because he was trying to persuade European allies to contribute their fair share to Ukraine’s defense against Russia.

“There was never any quid pro quo,” Mr. Trump said. “I said, ‘Hold it up. Let’s get other people to pay.’ I think that other countries should be paying, also. I wanted to get other countries.”

Mrs. Pelosi, though, said the president’s confirmation of the call was a “betrayal of his oath of office, betrayal of our national security and betrayal of the integrity of our elections.”

Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence, said the whistleblower at the center of the controversy is looking to testify before his committee “as soon as this week.” Although the committee was still working to finalize a hearing, he told reporters that it would be behind closed doors.

The Trump administration reportedly has dropped its objection to the whistleblower’s testimony.

In Wilmington, Delaware, Mr. Biden said Mr. Trump’s stonewalling would likely leave Democrats with no choice but to start impeachment proceedings. He said the impeachment of Mr. Trump “would be a tragedy of his own making.”

“This is not the conduct of an American president,” Mr. Biden said. “It is the abuse of power. It undermines our national security. It violates his oath of office, and it strikes at the heart of the sworn responsibility of the president.”

Other Democratic presidential candidates who were reluctant to call for impeachment proceedings reversed themselves Tuesday. They said they now believe Mr. Trump went too far by trying to pressure Ukraine.

On Capitol Hill, Democrats who were agitating to begin impeachment proceedings said Mrs. Pelosi’s announcement was a watershed moment.

“Speaker Pelosi is telling us that we’re full steam ahead and the country will be with us if they aren’t already,” Rep. Jamie Raskin, Maryland Democrat, told reporters.

It’s not clear what changes with Mrs. Pelosi’s announcement. House Judiciary Committee Chairman Jerrold Nadler, New York Democrat, has been telling courts for two months that he has been conducting an impeachment inquiry.

His committee is looking into Mr. Trump’s personal business dealings and retracing the trail of special counsel Robert Mueller, who said there could be reason to believe Mr. Trump tried to obstruct justice in the investigation into Russian meddling in the 2016 presidential election.

Mrs. Pelosi said she is directing Mr. Nadler and five other committee chairmen to continue their work under the “umbrella of impeachment inquiry.”

Rep. James A. Himes, Connecticut Democrat and a member of the intelligence committee — one of the six panels Mrs. Pelosi said is operating as part of the inquiry — said he figures they will forward all of their findings to Mr. Nadler, who as Judiciary Committee chairman would lead any impeachment proceedings.

Still, some lawmakers said the investigations aren’t enough. Based on what they have seen, they said, the House should skip the inquiry stage and draw up articles of impeachment.

Impeachment is the equivalent of a political indictment and can be approved with a majority vote in the House. If articles of impeachment are approved, then the Senate holds a trial. It takes a two-thirds vote in the Senate to convict and remove the president.

Senate Majority Leader Mitch McConnell, Kentucky Republican, signaled that the Senate isn’t about to follow Mrs. Pelosi’s lead. He said Democrats have made a “rush to judgment.”

“It simply confirms that House Democrats’ priority is not making life better for the American people but their nearly 3-year-old fixation on impeachment,” he said.

Public counts showed that an overwhelming majority of House Democrats support beginning impeachment, and polling shows Democratic voters are with them. Republicans and independents have consistently dismissed the idea of impeachment in surveys.

Democrats had been expecting that flashpoints, such as the release of the special counsel’s report this spring, Mr. Mueller’s testimony to Congress in July and testimony by former top Trump aides over the past few months, would sway the public, yet polling hasn’t changed.

A Reuters/Ipsos poll released Tuesday night showed that 37% of Americans think Mr. Trump should be impeached, down from 41% in the same survey a month ago. The polling was conducted after reports about the Ukraine controversy surfaced.

Mr. Trump on Tuesday insisted the real malfeasance belongs to Mr. Biden and his son, who earned about $3 million from the Ukrainian energy company. As vice president in 2016, Mr. Biden threatened Ukraine’s leaders that the U.S. would withhold $1 billion in loan guarantees unless they fired the nation’s top prosecutor, whose office had been investigating the gas company. Mr. Biden and other Obama administration officials said they sought the prosecutor’s removal for failing to pursue corruption, not to take any heat off Hunter Biden.

Mr. Trump said in a Twitter message that the transcript will show his call with Ukraine’s president was “very friendly and totally appropriate.”

“No pressure and, unlike Joe Biden and his son, NO quid pro quo!” the president tweeted. “This is nothing more than a continuation of the Greatest and most Destructive Witch Hunt of all time!”

Mr. Zelensky, who is scheduled to meet with Mr. Trump Wednesday on the sidelines of the U.N. conference, said Tuesday that he respects Mr. Trump and wants the unceasing support of the U.S.

Mr. Zelensky said he expects his meeting with Mr. Trump “will be very warm.”

The president pointed out that the money was released to Ukraine after a two-month wait.

“As far as withholding funds, they were paid,” Mr. Trump said. “They were fully paid. I want other countries to put up money. People called me and said, ‘Let it go,’ and I let it go.”

A search of the White House website doesn’t reveal any summaries, or readouts, of phone calls made by Mr. Trump with foreign leaders last summer in which he discussed further payments for Ukraine’s defense.

The president said Tuesday that, similar to his complaints about NATO members not paying their fair share for the common defense of Europe, other European allies should be contributing more to protect Ukraine from Russian aggression.

“We paid the money,” Mr. Trump said. “But very importantly, Germany, France, other countries should put up money. And that’s been my complaint from the beginning.”

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