WASHINGTON—The House narrowly approved opening an impeachment probe into President Biden on Wednesday, hours after his son, Hunter Biden, defied a congressional demand to testify on Capitol Hill, marking a sharp escalation in the battle between the White House and Republicans.

The House voted 221-212 along party lines to formally authorize Republicans’ impeachment probe, which party leaders initiated several months ago, hoping to add legal and political muscle to the investigation into whether the president had ties to his son’s overseas business dealings. While GOP lawmakers have obtained testimony that Joe Biden before becoming president occasionally met with his son’s business associates, they haven’t uncovered support for those claims or established that he profited from his family’s overseas endeavors.

Democrats criticized the vote as a political stunt and an effort to exact retribution on behalf of former President Trump, who was impeached twice by the House before being acquitted by the Senate. The White House and Democrats have said Republicans have failed to find any evidence of wrongdoing by President Biden that approaches an impeachable offense.

The White House weeks ago challenged House subpoenas and demands for transcribed interviews with Biden family members on the grounds that the existing impeachment probe, launched by GOP leaders in September, wasn’t valid because the House didn’t vote to authorize it.

Speaker Mike Johnson (R., La.) and other GOP leaders said in a joint statement that the vote “puts us in the strongest position” to enforce subpoenas in court and overcome what they called stonewalling by the White House. Rep. Jim Jordan (R., Ohio), chairman of the House Judiciary Committee, said the House has “now spoken, pretty loudly and pretty clearly” in backing an official impeachment inquiry.

President Biden said Republicans were “attacking me with lies” and “choosing to waste time on this baseless political stunt.”

Democratic lawmakers remained united behind the president.

“No amount of evidence could convince Republicans that President Biden did nothing wrong because they’re not looking for truth, they’re looking for revenge,” Rep. Jim McGovern of Massachusetts, the top Democrat on the House Rules Committee, said on the House floor ahead of the vote.

Impeachment investigations give Congress additional power by increasing the likelihood that a court would authorize access to grand-jury materials and improving the chances of overcoming assertions such as executive privilege. While there is disagreement over whether a formal House vote is needed to authorize an impeachment probe, such a move helps remove legal ambiguity.

The House Oversight Committee has been focused on the financial dealings of the president and his family, while the House Judiciary Committee has been investigating what Republicans portray as the weaponization of the Justice Department and the Federal Bureau of Investigation.

The vote Wednesday followed rare public remarks from Hunter Biden at the Capitol earlier in the day. He said he was willing to testify publicly, not behind closed doors, so that Republicans couldn’t selectively leak portions of his statements.

“I’m here today to make sure that the House committee’s illegitimate investigations of my family do not proceed on distortions, manipulated evidence and lies,” he said. “Let me state as clearly as I can: My father was not financially involved in my business.”

The younger Biden has faced congressional and legal scrutiny regarding his overseas business dealings in Ukraine, China and elsewhere as well as alleged tax evasion, and Republicans have sought to show links between Hunter’s work and his father.

Jordan said the private format better positions lawmakers to focus on the facts rather than encourage public performances.

White House press secretary Karine Jean-Pierre said that the president was aware of what his son planned to say at the Capitol. Jean-Pierre said his parents were “proud of him, continuing to rebuild his life.”

House Democrats and the White House have denied any effort to impede the investigation.

Wednesday’s vote ensures that the political controversy around the impeachment inquiry will extend into next year, as the pressures of an election year intensify. Vulnerable Republicans in competitive districts made clear that their support for Wednesday’s procedural step didn’t lock them into voting to impeach Biden at the end of the investigation.

“We need to have a formal inquiry to get the information. And I do not directly think this is going to lead to an impeachment,” said Rep. Don Bacon (R., Neb.), whose district was won by Biden in 2020. He said the current evidence suggested to him that Biden probably hadn’t committed offenses that could be considered a high crime or misdemeanor warranting impeachment.

Wednesday’s vote showed how far House Republican leaders have moved since early this year, when then-Speaker Kevin McCarthy (R., Calif.) resisted scheduling such a vote amid reluctance from moderates in competitive districts hoping to keep their seats in 2024. Instead, McCarthy said that he was directing House committees to open an impeachment inquiry, without a House vote.

While casting a future vote to impeach Biden could revive anxiety among Republicans in swing districts, a recent Wall Street Journal poll found essentially no effect on the presidential race if the House impeached Biden because of his involvement with his son’s foreign business affairs.

Trump led Biden in the survey by 4 percentage points, 47% to 43%. When voters were asked how they would vote if Biden were impeached, Trump then held a 5-point lead, 46% to 41%.

Trump has been publicly pressuring Republicans to move more swiftly on Biden. “Either IMPEACH the BUM, or fade into OBLIVION. THEY DID IT TO US!” he wrote in a late August social-media post. Aides to Trump say the subject didn’t come up when he met last month with Johnson during a fundraiser at Mar-a-Lago in Florida.

“There could be nothing there,” Rep. Dave Joyce (R., Ohio) said of the claims against Biden. “But there have been a lot of allegations made and I think it’s important people have the facts instead of allegations so they can make intelligent decisions.”

For his part, Hunter Biden faces both congressional scrutiny and growing legal peril.

Last week, he was charged with evading taxes on millions of dollars in income from foreign firms, in the second indictment against him in a span of four months. The younger Biden was previously indicted in September on felony charges alleging he lied about his drug use on a federal form he completed as part of a 2018 gun purchase.

The two indictments came in the fallout from the collapse in July of a plea deal in which Hunter Biden was set to admit to two misdemeanor tax offenses and avoid prosecution on a gun charge. On Monday, his lawyers urged a federal judge to dismiss the three gun charges, arguing that federal prosecutors had reneged on promises made in that plea agreement.

—Annie Linskey and Aaron Zitner contributed to this article.

Write to Kristina Peterson at kristina.peterson@wsj.com and C. Ryan Barber at ryan.barber@wsj.com