President Trump did not explicitly pressure the president of Ukraine for an investigation of Democratic front-runner Joseph R. Biden and his son in exchange for promised U.S. military aid, according to a government transcript released Wednesday, an account that the president said wrecks Democrats’ push for impeachment even as he girded for a long fight.

“It turned out to be a nothing call,” Mr. Trump said at the United Nations. “There was no pressure.”

But Mr. Trump did ask Ukrainian President Volodymyr Zelensky for an investigation of the Bidens during the July 25 phone call. That disclosure prompted House Democrats to harden their position on moving forward with impeachment. They said it is proof that Mr. Trump sought help from a foreign leader to interfere with a U.S. election.

The Justice Department’s release of the transcript unleashed another furious round of partisan accusations a day after House Speaker Nancy Pelosi, California Democrat, announced that her troops were opening a formal impeachment inquiry of the president.

Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence, said the details of the phone call were “far more damning” than he had imagined.

“What those notes reflect is a classic mafialike shakedown of a foreign leader,” Mr. Schiff said.

Mrs. Pelosi said after the release of the transcript that the president’s conduct “will not stand.”

At a press conference wrapping up three days of meetings at the United Nations, Mr. Trump expressed enormous frustration about another impeachment threat after the long-running drama of special counsel Robert Mueller’s Russia investigation.

“I thought we won,” the president said wistfully. “I thought it was dead.”

He spoke of former aides who had to pay legal fees in the Mueller investigation and said, “What these Democrats have done to ruin lives is so sad.”

He accused House Democrats of mounting the impeachment effort this week to obscure his administration’s “tremendous achievements” at the United Nations General Assembly.

“That was all planned, like everything else,” Mr. Trump said with bitterness in his voice.

The president said Mrs. Pelosi surrendered her authority in the name of extreme partisanship.

“She’s been taken over by the radical left,” Mr. Trump said. “People are really angry at the Democrats. Nancy Pelosi, unfortunately, as far as I’m concerned, she’s no longer the speaker of the House.”

He predicted that the impeachment drive will backfire and propel Republicans to take back the House next year.

“They’re in a very bad position,” he said of House Democrats. “I really think the Republicans will take over the House. Nancy Pelosi was forced [by liberal Democrats] into a position that she didn’t want to be in, and she wasn’t tough enough to stop it. A lot of her members now are having second thoughts.”

The Trump campaign said it raised $5 million in the first 24 hours after Mrs. Pelosi announced the impeachment inquiry. Several polls this week show public support for impeachment eroding, with at most one-third of voters in favor.

The Justice Department’s criminal division said it already had investigated Mr. Trump’s conversation with the Ukrainian president about Mr. Biden and concluded that Mr. Trump did not violate campaign finance laws.

A separate division of the department also ruled that the administration did not break the law by failing to quickly share a whistleblower’s complaint with Congress over the Ukraine matter, saying it didn’t meet the definition of “urgent” to trigger immediate action.

Mr. Trump said he authorized congressional Republicans to provide “full transparency” on the whistleblower’s report, which the House and Senate intelligence committees received Wednesday after weeks of stonewalling by the administration.

In a rare bipartisan move, the House passed a nonbinding resolution by a 421-0 vote to rebuke Mr. Trump for delaying the release of the whistleblower complaint and reprimanding the White House for its “highly inappropriate efforts” to discredit the complaint.

But in a predictable split on the complaint, Democrats who saw it late Wednesday said it corroborated their concerns while Republicans downplayed it.

Mr. Zelensky, who met with Mr. Trump on Wednesday at the United Nations, told reporters he didn’t feel pressured during their phone call this summer.

“Nobody pushed me,” Mr. Zelensky said in halting English as he sat next to Mr. Trump. “I don’t want to be involved [in] democratic open elections of USA. We had good phone call. It was normal.”

The president had his 30-minute phone call with the Ukrainian leader ostensibly to congratulate Mr. Zelensky on his party’s victories in parliamentary elections.

But Mr. Trump also broached the question of whether Mr. Biden pressured Ukraine in 2016 as vice president to stop the investigation of a Ukrainian gas company that had paid Hunter Biden $3 million.

“There’s a lot of talk about Biden’s son, that Biden stopped the prosecution and a lot of people want to find out about that, so whatever you can do with the Attorney General [William Barr] would be great,” Mr. Trump said on the call. “Biden went around bragging that he stopped the prosecution, so if you can look into it … It sounds horrible to me.”

He also asked Mr. Zelensky to work with his personal attorney, Rudolph W. Giuliani, as he dug into the origins of the Russia collusion investigation that dominated nearly three years of his presidency and found no conspiracy with Moscow.

Mr. Zelensky responded on the call that the country’s new prosecutor would be loyal to him.

“He or she will look into the situation, specifically to the company that you mentioned in this issue,” he pledged to Mr. Trump.

Mr. Biden said in a statement that the transcript shows Mr. Trump “implored the president of Ukraine to work with his personal attorney to manufacture a smear against a domestic political opponent.”

“He has put his own political interests over our national security interest, which is bolstering Ukraine against Russian pressure,” Mr. Biden said. “Congress must pursue the facts and quickly take prompt action to hold Donald Trump accountable.”

The president also defended Mr. Giuliani for poking around Ukraine in search of information related to the 2016 campaign, the missing emails of Democratic candidate Hillary Clinton and how the Russian meddling investigation enveloped the Trump campaign.

“Rudy’s looking to also find out where the phony witch hunt started, where it started,” Mr. Trump said of the former New York mayor. “Where did it start? How come it started?”

He said his team is still trying to learn what happened to Mrs. Clinton’s 33,000 deleted emails from her term as secretary of state.

“You can’t get rid of evidence like that,” the president said. “It’s corrupt government.”

Turning to Mr. Zelensky, Mr. Trump said, “We have corruption also, Mr. President.”

Asked by a reporter whether he still wants the Ukrainian leader to investigate the Bidens, Mr. Trump replied, “No, I want him to do whatever he can. This was not his fault; he wasn’t there [in 2016]. But whatever he can do in terms of corruption, because the corruption is massive.

“When Biden’s son walks away with millions of dollars from Ukraine, and he knows nothing, and they’re paying him millions of dollars, that’s corruption,” Mr. Trump said.

Asked how he would have felt if President Obama tried to dig up dirt on him from foreign leaders in 2016, Mr. Trump responded, “That’s what he did, really … how that whole witch hunt started.”

The two presidents bantered in front of reporters, joking about the timing of their meeting in the midst of the political firestorm.

“It’s better to be on TV than by phone,” Mr. Zelensky, a former comedian, told Mr. Trump to laughter.

During their call this summer, the two leaders also talked about the U.S. cybersecurity firm CrowdStrike and the controversy over the Democratic National Committee servers that were hacked in 2016. But no “quid pro quo” is evident in the written account of the phone conversation, which was compiled by national security aides who listened to the call and took notes.

Although there is no overt link to U.S. military aid, Mr. Trump does ask Mr. Zelensky for a “favor” immediately after reminding the Ukrainian leader that the U.S. has spent “a lot of effort and a lot of time” on Ukraine.

“I would like you to do us a favor though because our country has been through a lot and Ukraine knows a lot about it,” Mr. Trump said on the call. “I would like you to find out what happened with this whole situation with Ukraine, they say Crowdstrike. … I guess you have one of your wealthy people. … The [DNC] server, they say Ukraine has it. There are a lot of things that went on the whole situation. … I would like to have the attorney general call you or your people and I would like you to get to the bottom of it.”

The DNC hired CrowdStrike initially to examine its servers in the Russian hacking affair. The DNC never turned over the servers to the FBI.

The president also mentions the recently completed investigation of special counsel Robert Mueller over unfounded allegations of Russian collusion with the Trump campaign. Mr. Trump and Mr. Giuliani believe the “witch hunt” investigation had its origins in Ukraine.

“They say a lot of it started with Ukraine,” Mr. Trump told Mr. Zelensky. “Whatever you can do, it’s very important that you do it if that’s possible.”

Mr. Trump said the U.S. is helping Ukraine “much more than the European countries are doing, and they should be helping you more than they are.”

“Germany does almost nothing for you,” Mr. Trump said. “The United States has been very, very good to Ukraine. I wouldn’t say that it’s reciprocal necessarily because things are happening that are not good, but the United States has been very, very good to Ukraine.”

Mr. Zelensky replied, “You are absolutely right. I’m very grateful to you for that because the United States is doing quite a lot for Ukraine. Much more than the European Union especially, when we are talking about sanctions against the Russian Federation.”

House Democrats announced a formal impeachment inquiry Tuesday based on their suspicions that Mr. Trump tried to pressure the Ukrainian leader into investigating the Bidens by withholding $391 million in military aid last summer. Mrs. Pelosi said Mr. Trump had committed a “betrayal” of the Constitution.

Congressional Republicans said the transcript of the Zelensky call proves the president did nothing wrong.

Sen. Lindsey Graham, South Carolina Republican, said “I don’t mind” that Mr. Trump raised concerns about the Bidens in the call.

“I think it’s very appropriate for the president of the United States to suggest that you’ve got a corruption problem,” Mr. Graham told reporters. “Did the president of the United States suggest to the Ukraine, ‘I will withhold money unless you go after my political rival’? The answer is absolutely not. That’s why I wanted the phone call to be released.”

Mr. Graham said Mr. Biden’s son “was receiving money from the Ukraine, was on the board of a company that was the subject of an investigation.”

“If you don’t see that conflict, you’re blind,” he said.

Sen. Mitt Romney, Utah Republican and the party’s 2012 nominee for president, said the transcript was “deeply troubling.”

A whistleblower reported the conversation to authorities Aug. 12, citing information from “White House officials” and saying it appeared Mr. Trump was trying to pressure Ukraine to assist in his 2020 election.

The intelligence community inspector general concluded that the whistleblower showed political bias but said the report was credible and urgent enough that the director of national intelligence needed to share it with Congress.

But the Justice Department’s office of legal counsel said that was not the case.

“The question is whether such a complaint falls within the statutory definition of ‘urgent concern’ that the law requires the DNI to forward to the intelligence committees. We conclude that it does not,” wrote Steven A. Engel, assistant attorney general at the office of legal counsel.

They also ruled that the president’s conversation was a diplomatic communication and not an intelligence activity and the president isn’t a part of the intelligence community anyway, so his behavior is not part of the DNI’s purview.

“Such matters simply do not relate to ‘the funding, administration or operation of an intelligence activity within the responsibility and authority’ of the DNI,” Mr. Engel concluded.

Mr. Trump said he “hated” to release the transcript of the phone call but felt there was no other way to address the accusations of wrongdoing.

“I don’t like the concept of releasing calls. It’s hard to do business that way,” Mr. Trump told reporters. “But you folks [in the media] were saying such lies.”

He added, “I used to be the king of getting good press. [But] if you see the way they treat my family … It’s so bad for our country.”

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

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