This story was updated on Aug. 6, 2015.

Republicans are hoping that revulsion over a series of stealth videos featuring Planned Parenthood will swing the politics of abortion in their favor.

On Monday, the Senate took up a GOP bill to strip the women’s health-care provider of its $500 million in annual federal funding. Democrats blocked the bill from moving forward, but it was a chance for Republicans to go on record with their support for the idea.

Conservative Republicans warn they could attach a defunding rider to must-pass legislation in the fall, threatening a government shutdown. Senate majority leader Mitch McConnell (R) of Kentucky, however, states "there will be no government shutdown."

Social conservatives, marginalized over gay marriage, have been looking for every opportunity they can to put forward their antiabortion agenda, a core value for them. They’ve made steady inroads at the state level, bringing the number of enacted state restrictions on abortion to 282 in the past five years, according to the Guttmacher Institute.

But nationally, abortion as a GOP issue has often backfired in a country that is consistently split on the issue.

Republicans are “latching on to the videos because they are very disturbing for anybody,” especially their base, explains Dianne Bystrom, director of the Carrie Chapman Catt Center for Women and Politics at Iowa State University in Ames. The series of sting videos shows jarring visuals and Planned Parenthood employees casually discussing the sale of aborted fetal-tissue parts for medical research.

The current GOP effort “reminds me of the partial-birth abortion debate,” she says. But, she adds, “It’s going to be tricky for them.”

Nearly 20 years ago, graphic descriptions and depictions of “partial-birth abortion” roiled Congress and the presidential election. The gruesomeness was so powerful that it eventually led to a federal ban on this type of late-term abortion in 2003. The US Supreme Court upheld the ban in 2007.

More recently, the GOP has hit some roadblocks in its efforts to restrict abortion. In 2011, Republicans tried – and failed – to defund Planned Parenthood. In 2012, comments about “legitimate rape” by a GOP Senate candidate appalled many women voters, and President Obama won reelection as Democrats pointed to a GOP “war on women.”

Earlier this year, the House Republican leadership had to pull back a bill that banned late-term abortions because its own female members complained that the legislation didn’t allow exceptions for rape and incest.

“By and large, most people support Planned Parenthood,” Ms. Bystrom says.

Only a small percentage of Planned Parenthood’s services involve abortion, and those are not supported by tax dollars. The group provides a wide variety of preventive services, from birth control to screenings for disease. One in 5 women has relied on Planned Parenthood for health care in her lifetime, and many of its patients are low- and middle-income women.

According to a poll carried out for Planned Parenthood by Democratic pollster Hart Research Associates, 63 percent of registered voters oppose cutting off funding for the health-care provider. The poll was taken more than a week after the first video was released by the Center for Medical Progress, an antiabortion group.

In a Washington Post opinion piece, Planned Parenthood president Cecile Richards called the video claims of selling parts for profit “false and absurd,” though she quickly apologized for the tone of comments that were recorded. It is not illegal to pass on aborted fetal tissue for medical research or to have those costs covered, though trafficking is illegal.

“Attacking this funding is attacking women who need preventive health care,” Ms. Richards wrote.

Republicans, well aware of the trickiness of the issue, sought to make the Senate defunding bill more palatable by diverting the Planned Parenthood funds to other women’s health organizations that don’t perform abortions. Also, the bill was sponsored by a woman, freshman Sen. Joni Ernst (R) of Iowa, who ran an unabashedly antiabortion campaign last year.

“As a mother and a grandmother, the gravity of Planned Parenthood’s callous and morally reprehensible behavior cannot be ignored,” she said in a floor speech Monday.

But the moderate Sen. Susan Collins (R) of Maine voiced concerns that other health centers would be overwhelmed if they had to pick up Planned Parenthood patients – a point vigorously made by Democrats. Senator Collins supported an amendment with fellow Republican Mark Kirk of Illinois to have the Department of Justice investigate whether Planned Parenthood had engaged in any wrongdoing with the fetuses.

Political analyst Kyle Kondik points out that while the “war on women” strategy worked for Democrats in 2012, it didn’t in 2014 – with the election of Senator Ernst as a good example.

“Democrats in recent years have placed a very heavy focus on using reproductive rights issues as a way to attack Republicans. There’s a sense among some that the Democrats have taken it too far,” he says.

Mr. Kondik, with the University of Virginia Center for Politics, wonders how much traction the issue is even getting nationally. Indeed, 52 percent of Americans are not following the controversy closely, according to a Christian Science Monitor/TIPP poll that was conducted July 25-30. Americans are paying much more attention to the Iran nuclear agreement and Hillary Clinton’s e-mails than they are to the Planned Parenthood videos, according to the poll.

Of those who are closely following the issue (60 percent of Republicans, 46 percent of independents, and 36 percent of Democrats), a majority – 58 percent – say that funding for Planned Parenthood should be cut off.

Still, Raghavan Mayur, president of TechnoMetrica Market Intelligence, which conducts the Monitor/TIPP poll, does not think GOP leaders in Congress will go for a government shutdown over the issue.

“It’s not going to escalate like Obamacare,” says Mr. Mayur, referring to the 2013 partial-government shutdown over the Affordable Care Act. “I don’t think the country is in the mood for that kind of thing.”

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Meanwhile, Mr. Obama has said he would veto any bill defunding the women’s health-care provider.

But that won’t stop the Planned Parenthood issue from being used as a political rallying cry. Both parties are already fundraising over it.