WASHINGTON—The House late Tuesday passed legislation that would represent the most significant change to labor law in decades, advancing a priority for unions that are pressuring President Biden and Democrats to deliver legislative victories.

The Protecting the Right to Organize Act, or PRO Act, was approved by the House 225-206, with the support of nearly all Democrats and five Republicans. Backers say the legislation would be a major advancement in employee rights following recent setbacks for organized labor and diminished union membership.

Business groups such as the U.S. Chamber of Commerce and the App-Based Work Alliance, which includes Uber Technologies Inc., Lyft Inc. and DoorDash Inc., assailed the bill, saying it would trample on state laws and endanger the flexibility that ride-share and delivery drivers enjoy.

Mr. Biden called on Congress to send the legislation for his signature, but it faces a steep challenge in the evenly divided Senate. Progressive Democrats want Mr. Biden to endorse eliminating the Senate’s 60-vote threshold for most legislation, a change he and some centrists in his party oppose. There is also no guarantee all 50 Democratic senators would support the union bill.

Labor groups are looking to Mr. Biden to push for the bill as the latest sign he is delivering on his campaign promise to boost unions. The president has fired officials at the National Labor Relations Board seen by unions as hostile to their goals, sided with Amazon.com Inc. workers’ unionization drive in Alabama, picked a union ally to run the Labor Department and backed increasing the federal minimum wage to $15 per hour, though that effort is currently stalled.

The PRO Act would establish penalties up to $50,000 for employers who violate the National Labor Relations Act and set a 10-day timeline for union-employer negotiations to commence; currently there is no enforceable timeline.

The bill also makes it easier for gig workers—including independent contractors and online platform workers—to unionize unless their employer meets certain requirements. It would codify into law a decision made by the National Labor Relations Board under former President Barack Obama and reversed under former President Donald Trump that could have paved the way for contractors and workers at franchised businesses to form unions.

“The PRO Act is our litmus test and if progress is delayed or denied yet again, the suffering of the past year will only get worse,” AFL-CIO President Richard Trumka said, referring to the coronavirus pandemic. “But if our leaders step up to the plate and deliver generational change—the change we voted for—we will emerge from this crisis stronger than ever before.”

Companies such as Uber and Lyft won a victory against labor groups last year in California, after spending $200 million in support of a ballot measure allowing them to bypass a state law intended to provide employee-like protections for drivers. Now, despite low Senate prospects, the industry is working to defeat the federal legislation.

The App-Based Work Alliance, the industry group, on Tuesday said the House “voted to put at risk the flexibility that millions of app-based workers rely on by including the same overly narrow definition of independent work” as the California law. The Chamber of Commerce said that the bill would upend right-to-work laws passed in 27 states and that the group would run TV ads in opposition.

Many Republicans say the House bill will stifle business and empower union leaders, and some key moderate Democrats haven’t yet weighed in. The legislation could draw more attention to the brewing fight over the Senate filibuster, the requirement that most legislation have 60 votes to pass, because Democrats only have 51 votes if they stick together.

Mr. Biden is already facing some criticism from progressive Democrats and some in organized labor who say he should have fought harder to include the minimum wage increase in his $1.9 trillion pandemic relief package. The White House and centrist Democrats disagreed with calls from the left to overrule a Senate parliamentarian ruling that the wage increase is outside of the scope of the budget maneuver Democrats used to pass the relief bill without GOP support.

“The administration talked a good game about $15, but they failed to deliver,” said Joseph Geevarghese, executive director of the progressive group Our Revolution, which held a rally outside the Capitol on Tuesday in support of the PRO Act. “The president can do much more.”

Mr. Biden opened his 2020 campaign at a union hall in Pittsburgh and, near the end of the run, pledged to be the most pro-labor president. The message was in part aimed at rank-and-file union members who have shifted support to Republicans, particularly Mr. Trump, even as union leadership supports Democrats. The candidates battled for the union vote across several Midwestern states crucial to Mr. Trump’s 2016 win.

In the 2020 cycle, some union members knocked doors in support of Democratic candidates even while many of the campaigns were still focused on virtual campaigning because of the pandemic. Democratic candidates in the election received more than 88% of the $79 million donated by labor PACs and union-affiliated individuals, according to the nonpartisan Center for Responsive Politics.

Union membership fell to 10.8% of the overall workforce in 2020, down from its recorded peak of 20.1% in 1983, according to the Labor Department, due to factors including corporate resistance and a shift from manufacturing. Some public opinion polls, though, show rising support for unions since the 2009 recession.

Labor leaders were disappointed by a lack of major victories under Mr. Obama. And some of Mr. Biden’s moves as president have drawn complaints, such as derailing the Keystone XL oil pipeline, which unions said would create 10,000 jobs, though many would have been temporary. His support for reducing fracking and other energy policies could also put him at odds with unions. The White House says Mr. Biden’s push for clean-energy projects will spur longer lasting job growth.

Mr. Biden’s push to open schools in the pandemic also has put him in the middle of a conflict between some parents and officials who want to quickly return to in-person learning and teachers’ unions, some of which want members vaccinated or other steps in place before fully reopening.

The White House says Mr. Biden is committed to working with lawmakers to raise the minimum wage, pass the PRO Act and move another economic package, which is expected to include infrastructure spending that could benefit union workers. During his campaign, Mr. Biden proposed requiring companies receiving government contracts to commit to paying at least $15 an hour and guarantee a choice to join a union and bargain collectively.

Democrats are discussing using reconciliation—the budget maneuver they used to pass the $1.9 trillion Covid measure with no Republican votes—for the second economic package. But Sen. Joe Manchin (D., W.Va.), one of his party’s most prominent centrists, has expressed concern about another party line legislative push.

*By Alex Leary and Eliza Collins, Wall Street Journal*