WASHINGTON — Senate Democrats suffered what looked like a difficult setback on birth control Wednesday, but they hope it pays big political dividends in November.
Republicans blocked a bill that was designed to override a Supreme Court ruling and ensure access to contraception for women who get their health insurance from companies with religious objections. The vote was 56-43 to move ahead on the legislation — dubbed the ‘‘Not My Boss’ Business Act’’ by proponents — four short of the 60 necessary to proceed.
But Democrats hope the issue has enough life to energize female voters in the fall, when Republicans are threatening to take control of the Senate.
GOP senators said Wednesday’s vote was simply a stunt, political messaging designed to boost vulnerable Democratic incumbents. The GOP needs to gain six seats to seize control.
‘‘Democrats are just trying to win an election,’’ said Senator Lamar Alexander, a Tennessee Republican.
But Senate majority leader Harry Reid, a Nevada Democrat, said the Republicans were the ones ‘‘out of touch with reality.’’ He promised that Democrats would continue to press the issue.
Women have proved crucial in electing President Obama and members of his party. And Democrats desperately need a strong turnout as they defend 21 Senate seats to the GOP’s 15, many in Republican-leaning states where Obama’s abysmal approval ratings are a probable drag.
Last month, the Supreme Court ruled 5-4 that requiring closely held companies to pay for various forms of women’s contraception to which they object violates the corporations’ religious freedom. The decision marked the first time the high court had declared that businesses can hold religious views under federal law.
‘‘Five men on the Supreme Court rolled back the clock on women in America,’’ said Senator Patty Murray, a Democrat from Washington state.
A Senate veteran — the four-term Murray — and an incumbent locked in a tight race — Colorado’s Mark Udall — joined forces in pushing the legislation that would have reversed the court’s ruling by providing access to contraception through insurance plans at businesses that object on religious grounds.
Republicans asserted that the government must accommodate the deeply held religious beliefs of Americans, including the owners of Hobby Lobby, the Oklahoma-based chain of arts and craft stores that challenged the contraceptives provision in the law.
‘‘The issue in Hobby Lobby is not whether women can purchase birth control, it’s who pays for what,’’ said Senator Deb Fischer, a Republican from Nebraska, in remarks on the Senate floor. ‘‘Those of us who believe that life begins at conception have moral objections to devices or procedures that destroy fertilized embryos.’’
Fischer said the Green family, which owns Hobby Lobby, has similar objections and ‘‘they don’t want to use their money to violate their religious beliefs.’’ She said the company’s health coverage does pay for 16 of 20 forms of contraception, including birth control pills.
Senate Minority Leader Mitch McConnell, a Kentucky Republican, said Democrats ‘‘think they can score political points and create divisions where there aren’t any by distorting the facts.’’
McConnell joined with two Republican women, Fischer and Senator Kelly Ayotte of New Hampshire, in backing separate legislation that would reaffirm current law on access to contraception and in calling for a Food and Drug Administration study on whether contraceptives could be sold over the counter without a prescription.
In one of the most closely watched races in the country, McConnell faces Democrat Alison Lundergan Grimes in his bid for a sixth term.
On Wednesday’s vote, three Republicans broke ranks with their party — Susan Collins of Maine, Lisa Murkowski of Alaska, and Mark Kirk of Illinois — and backed the Democratic-led legislation. In a procedural move, Reid switched his vote to no, allowing him to bring the measure up for another vote closer to the election.
All other Democrats backed the bill.
Democrats facing reelection insisted that the court ruling would force some women to pay out of pocket for contraceptives, or simply skip the purchase if the cost was too much.
‘‘When you charge women more for contraceptive coverage, then you are denying them access to that care,’’ said Senator Jeanne Shaheen, a New Hampshire Democrat, who is in a competitive race.
The government has said nearly 30 million women get birth control as a result of the four-year-old health care law.