Mr. Menendez said in a statement that he would consider giving the name back to the Republicans on one condition.

“As soon as they come up with a bill that actually cuts taxes for the middle class and creates jobs, they can have it back,” he said.

House takes a victory lap

House Speaker Paul D. Ryan of Wisconsin, speaking on the House floor, called the vote “a turning point” saying “this is our chance, this is our moment.” When the bill passed the House, a giddy Mr. Ryan smiled broadly and banged the gavel with force as he declared victory.

Passage of the bill came over the strenuous objections of Democrats in both the House and the Senate, who have accused Republicans of giving a gift to corporations and the wealthy and driving up the federal debt in the process. The final vote tally can be found here.

Twelve House Republicans also voted no on the bill, including lawmakers from high-tax states like New York, New Jersey and California.

Under the bill approved by the House, the corporate tax rate would be cut to 21 percent, from the current 35 percent, a move that Republicans are betting will increase economic growth. Individuals would also see tax cuts, including a top rate of 37 percent, down from 39.6 percent. But the individual tax cuts would expire after 2025, a step that Republicans took to comply with budget rules that allow them to pass the tax overhaul using a simple majority, and without Democratic votes.

Republicans in Congress moved with remarkable speed to enact the biggest tax overhaul since 1986, unveiling legislation to rewrite the tax code, marshaling support for their effort and devising a compromise between the House and Senate in under two months.