It is “unlikely” that either measure will be taken up in the Republican-controlled Senate, Senator John Thune of South Dakota, the No. 3 Republican, said on Tuesday. House Republicans have flatly condemned both pieces of legislation, arguing the measures would be both ineffective and infringe on law-abiding citizens’ rights.

The House bill that was approved on Wednesday “is taking the fears and concerns of a nation over gun violence and perpetrating a fraud on them,” said Representative Doug Collins of Georgia, the top Republican on the Judiciary Committee. “The reality is nothing in this bill would have stopped Parkland, or stopped any of the other violence we have seen.”

Gunmen at the center of a number of high-profile mass shootings passed background checks and legally purchased the firearms they used. But gun control activists have praised both background check measures as a critical first step in a broader drive to reduce gun violence.

“America’s current background check system is like having two types of security lines at the airport: one for people who are willing to be screened, and one you can waltz right through carrying whatever you want,” said John Feinblatt, the president of Everytown for Gun Safety. “We applaud the House for moving so quickly.”

Once divided on the issue, House Democrats returned to the majority, promising meaningful action on gun control, with aid from advocacy groups backing candidates who made the issue central to their campaigns. Michael Bloomberg, the former mayor of New York, spent around $100 million in the midterms last year to elect gun control supporters, according to Everytown.

The group will continue its push this week with $400,000 in digital ads urging lawmakers to support the legislation as it tries to break through the Senate.

One of the new freshman Democrats supported by Mr. Bloomberg, Representative Lucy McBath of Georgia, had a 17-year-old son, Jordan, killed by gun violence.