After a hastily called meeting on Tuesday morning among Republicans, the matter was dropped before it could go to the full House floor for a vote.

As the Senate moved to larger legislative matters, the House kerfuffle seemed to cast a shadow over Mr. Ryan, but he tried to brush it off. “There’s no sense of foreboding in the House today,” Mr. Ryan said after his re-election, “only the sense of potential.”

The fight over the House rules was already acrimonious thanks to a piece of the package that would impose $2,500 in fines for filming events on the House floor, a response to Democrats who streamed their overnight sit-in over guns last June using cellphones and video cameras.

In the Senate, Vice President Joseph R. Biden Jr. swore in seven new members and all the incumbents who won their races last year, their colleagues looking on cheerfully, as a cold rain pelted the newly refurbished Capitol dome.

Members of the House and Senate brought along their families — elderly parents with canes, small children tugging at uncomfortable lacy hems — as well as former senators and other special guests. Former Vice President Dick Cheney accompanied his daughter Liz to her swearing-in as a member of the House elected from Wyoming.

Senator Chuck Schumer of New York officially became the Democratic leader and quickly warned Republicans that the minority would be vocal, if not operatic, in resisting much of their agenda and many of Mr. Trump’s nominees.

“It is our job to do what’s best for the American people, the middle class and those struggling to get there,” he said. “If the president-elect proposes legislation on issues like infrastructure, trade and closing the carried interest loophole, for instance, we will work in good faith to perfect and, potentially, enact it. When he doesn’t, we will resist.”