WASHINGTON — With much of the Democratic Party raging over the seemingly imminent end to the almost 50-year-old constitutional right to an abortion, Senator Chuck Schumer of New York vowed on Tuesday to bring legislation to a vote that would codify a woman’s right to end her pregnancy.

But Mr. Schumer, the majority leader, and fellow Democrats in Washington appeared to find themselves powerless to stop the overturning of Roe v. Wade, as the battle for abortion rights shifted from Washington to the states.

The House voted 218 to 211 in September to codify the right to an abortion at the federal level after a Texas law made almost all abortions illegal and deputized individual citizens to enforce the prohibition. But in an evenly divided Senate, Democrats have neither the 60 votes necessary to bring up the House bill for consideration nor the 50 votes they would need to change the filibuster rules and allow it to pass with a simple majority.

Even if they did, Democrats are likely short of a simple majority in favor of abortion-rights legislation. Senator Joe Manchin III, the conservative-leaning Democrat from West Virginia, joined Republicans in February to block consideration of the House-passed bill when Mr. Schumer tried to bring it up the first time.

Two Senate Republicans, Susan Collins of Maine and Lisa Murkowski of Alaska, do support abortion rights, and Ms. Collins expressed frustration on Tuesday that the draft decision overturning Roe was “completely inconsistent” with the assurances that Justices Brett M. Kavanaugh and Neil M. Gorsuch had given her about respecting precedent and Roe being the law of the land.

But neither Republican has been willing to break with her leaders and join Democrats in a bid to put an end to the filibuster, leaving Democrats with no path to bringing up a bill to enshrine Roe into federal law.

Democrats were left steaming in remarkably vociferous language. Mr. Schumer said the conservative justices who had promised to be faithful to legal precedent had “lied to the U.S. Senate, ripped up the Constitution and defiled both precedent and the Supreme Court’s reputation, all at the expense of tens of millions of women who could soon be stripped of their bodily autonomy.”

Speaker Nancy Pelosi of California called the draft decision “monstrous” and a blueprint “for Republicans to obliterate even more of our freedoms” because denying a constitutional right of privacy would make other rights, including same-sex marriage and contraception, vulnerable to attacks from social conservatives.

In contrast, Republicans were largely subdued in their celebration, more focused on how the Supreme Court’s draft decision came to light than the decision itself.

“Yesterday’s unprecedented leak is an attempt to severely damage the Supreme Court,” House Republican leaders wrote in a joint statement, asserting without evidence that the draft decision overturning Roe v. Wade obtained by Politico was part of a “clearly coordinated campaign to intimidate and obstruct the Justices of the United States Supreme Court.”

Almost as an afterthought, the statement concluded that the leaders were praying “for a decision that protects our most basic and precious right, the right to life.”

Senator Mitch McConnell of Kentucky, the Republican leader who is normally quick to claim credit for the critical role he played in installing the conservative majority at the Supreme Court, was making no victory laps on Tuesday during his speech on the Senate floor.

Instead, he accused Democrats of smearing the justices and attempting to intimidate them by mobilizing “politicians, pundits, bullies and mobs.” The end of Roe seemed beside the point of his remarks.

The reactions suggested that at least some Republicans were wary of prompting a political backlash among voters to a ruling that is certain to thrill their conservative core supporters but could alienate the broad swath of Americans who support at least some form of abortion rights.

Democrats in Washington have few options left. Mr. McConnell’s 2016 blockade of President Barack Obama’s Supreme Court nominee, Merrick B. Garland, and the rapid replacement of Justice Ruth Bader Ginsburg with Justice Amy Coney Barrett just days before the 2020 election ushered in a six-justice conservative bloc that likely will not be dislodged for years.

The draft decision written by Justice Samuel Alito would send the question of abortion rights to state legislatures, and in the populous Northeast and West Coast, abortion would likely remain legal.

Even if Democrats could somehow eliminate the filibuster and muster 51 votes to codify Roe, the move could backfire; a Republican majority in Congress and a Republican president could then ban abortion nationwide with 51 votes — with no need to change Senate rules that Republicans have maintained are sacrosanct.

Democratic campaign officials immediately turned to the fall elections with the hope that the impending end of Roe v. Wade would energize young voters and turn Republican-leaning women in the suburbs against the G.O.P.

In a show of unity, the Democratic National Committee and the party’s campaign committees for the House, Senate, governors, state legislators and state attorneys general released a joint statement saying they were standing “shoulder to shoulder” to “elect Democrats who will serve as the last lines of defense.”

An end to constitutional abortion guarantees would raise the political stakes most in governors’ races and state legislative elections, where the future of abortion will be on the ballot.

But it is far from clear whether the issue will take precedence over inflation and economic struggles. About six in 10 Americans have said for decades that abortion should remain legal in all or most cases, said Amy Walter, a veteran congressional analyst at the nonpartisan Cook Political Report.

But what that means is not all that clear. Given four choices — legal under any circumstance, most circumstances, only a few cases or illegal in all cases — 45 percent of respondents told Gallup last year they would keep abortion legal under any or most circumstances, while 52 percent said they wanted it legal only in a few circumstances or illegal always, Ms. Walter noted.

The economy has been by far the most important issue in public opinion surveys. And abortion’s ability to mobilize voters is largely untested.