Speaker Mike Johnson on Tuesday tried and failed to kill a bipartisan effort to change House rules so that new lawmakers would temporarily be allowed to vote remotely after the birth of a child, suffering an embarrassing defeat that signaled that the proposal could soon be adopted.

In using strong-arm tactics to try to block the measure, Mr. Johnson was attempting an extraordinary use of the speaker’s power to prevent the chamber from even considering a measure backed by half its members. But he failed to peel off enough Republican votes to block the proposal, instead receiving a public rebuke at the hands of members of both parties.

The showdown on the House floor was a capstone of a long-running fight over the rights of new parents in Congress.

The saga began over a year ago, when Representative Anna Paulina Luna, Republican of Florida, began agitating for a change to House rules that would allow new mothers to designate a colleague to vote by proxy on their behalf for up to six weeks after giving birth. Ms. Luna landed on the idea after her own child was born.

There is no maternity or paternity leave for members of Congress, who can take time away from the office without sacrificing their pay but cannot vote if they are not physically in the Capitol. Proponents of the change have called it a common-sense fix to modernize Congress, where more women and more younger members serve now than did 200 years ago.