ALBANY—One of the most politically volatile issues of the legislative session—a move by Gov. Andrew Cuomo to amend state law regarding abortion rights—suffered defeat in the final hours of the New York state Senate proceedings Friday after a legislative leader attempted a last-ditch effort to advance it.

Sen. Jeffrey D. Klein—the leader of a breakaway group of Democrats who had been accused for weeks by Mr. Cuomo, a Democrat, of preventing a vote on the matter—introduced the measure as a hostile amendment to another bill already being debated on the floor.

Referencing a vote Tuesday in the Republican-led U.S. House of Representatives to impose the strictest new limits on abortion in a decade, Mr. Klein described the matter as "no longer about a political litmus test."

Because of a procedural quirk, however, the state Senate never took a vote on the actual amendment. Rather, 32 senators, including one Democrat, Sen. Ruben Diaz Sr. , overruled the chair on whether the amendment was "germane" to the bill, rejecting it before a vote could occur.

Mr. Klein, who is the co-leader of the Senate Majority Coalition alongside Sen. Dean Skelos, a Republican, had resisted introducing a package of bills proposed by Mr. Cuomo and aimed at women's rights, arguing that there weren't enough votes in the Senate to pass it. Mr. Skelos, meanwhile, has been steadfastly opposed to the abortion provision, with his spokeswoman having called it a "political maneuver" on the governor's part.

The bill, advocated by Mr. Cuomo since his January State of the State address, would change state law to allow late-term abortions when they would protect a woman's health, the same as federal law. The current state law says such abortions are allowed only when a woman's life is in danger, but it is superseded by the federal standard.

"I think it was clear that the other side of the aisle, my governing partners didn't want to bring this amendment to the floor," Mr. Klein said after the amendment's rejection. "And we thought the women of New York deserved a vote. And they got a vote today—not the right vote, but they got a vote."

Addressing whether the matter had frayed the fabric of the power-sharing coalition governing the Senate, Mr. Klein said that he had given Mr. Skelos "a heads up," adding that "on this specific issue, on the choice issue of upholding Roe v. Wade, we just couldn't come to an agreement with the state Senate Republicans, and it's unfortunate."

Mr. Cuomo's proposed legislation, originally introduced as an omnibus 10-part bill and broken up late Tuesday evening into 10 individual bills, also includes provisions aimed at pay equity and human trafficking. The state Assembly passed the omnibus package Thursday, while the senate passed nine of the bills, save the abortion piece, late Friday. The nine would need to be passed as individual bills by the Assembly to become law; in a statement Mr. Skelos said it was "unconscionable" for Assembly Speaker Sheldon Silver to fail to do so.

Mr. Silver, meanwhile, pinned the blame on the other chamber, saying: "We hope our partners in government will join us in finally passing the complete Women's Equality Act that includes protecting a woman's right to choose."

Write to Erica Orden at