WASHINGTON — Senator Chuck Schumer, Democrat of New York and the majority leader, said Thursday that he planned to move on Wednesday to bring up a bill that would codify abortion rights into federal law, moving quickly in the wake of a leaked Supreme Court decision overturning Roe v. Wade, despite clear evidence that the measure lacks the support to be enacted.

The move is little more than an effort to send a political message ahead of midterm elections and a seismic ruling that could have major legal, cultural and electoral consequences that also carries deep significance for voters across the political spectrum.

The legislation is all but certain to be blocked by Republicans, falling short of the 60 votes that would be needed to advance past a filibuster. It also appears to lack even the simple majority it would need to pass the 50-50 Senate, given that Senator Joe Manchin III, the centrist Democrat from West Virginia who opposes abortion rights, voted against bringing up a nearly identical measure in February and has shown no signs that he has shifted his position.

Even if Mr. Manchin did change his mind on the bill, he has adamantly opposed altering Senate rules to eliminate the filibuster, leaving Democrats short of the 50 votes they would need to do so and get their measure past a Republican blockade.

Still, Mr. Schumer said the vote next week would be one of “the most important we ever take,” framing it as an opportunity to emphasize to voters — who polls show widely favor at least some legal abortion — that elections matter, and that Democrats are the ones fighting to preserve reproductive rights.

“Senate Republicans spent years packing our courts with right-wing judges,” Mr. Schumer said in a speech on the Senate floor. “Will they now own up to the harm they’ve caused, or will they try to undo the damage? The vote next week will tell.”

He added: “Republicans can run but they can’t hide from the damage they’ve created.”

Even if Democrats have no real path to passing a bill to enshrine Roe into federal law, the vote will give them a chance to show their progressive core supporters that they are trying to do so. They also hope the action stokes a backlash against Republicans by swing voters, including college-educated suburban women, who might be alienated by the G.O.P.’s opposition to abortion rights.

Senator Mitch McConnell, Republican of Kentucky and the minority leader, and other top Republicans have mostly refrained from boasting about the impending demise of Roe since the draft opinion surfaced, focusing instead on the unprecedented Supreme Court leak. Their responses suggest that they, too, see the potential for a battle over abortion rights to hurt their party ahead of midterm congressional elections, and are working to reframe the issue to their advantage by portraying Democrats as extreme on the issue.

Democrats said their bill has gained urgency since the last time they tried to take it up. Back then, the threat to abortion rights was more theoretical. Now, they said, it has taken on new significance with the end to a constitutional right suddenly imminent.

They have also altered the measure in an effort to garner more support among pro-abortion-rights Republicans, removing a lengthy series of findings, including passages that referred to abortion restrictions as “a tool of gender oppression” and as being “rooted in misogyny.” Also scrapped was a section clarifying that while the bill references women, it is meant to protect the rights of “every person capable of becoming pregnant,” including transgender men and nonbinary individuals.

But the fundamentals of the bill, which states that medical workers can provide abortions to their patients, remain the same.

Democrats had hoped that removing such language could win over Republican Senators Susan Collins of Maine and Lisa Murkowski of Alaska, who both support abortion rights.

But Ms. Collins said on Thursday that she still opposes the bill, because it goes beyond simply codifying Roe v. Wade, which she supports, and lacks provisions that would allow Catholic hospitals to refuse to perform abortions.