Washington (CNN) Members of the House of Representatives have no say in whether professor Christine Blasey Ford's accusations against Judge Brett Kavanaugh sink his Supreme Court nomination.

But as much as the contentious confirmation fight over Kavanaugh played out on television screens across the country on Thursday has put vulnerable Democratic senators in the spotlight, it may be at-risk House Republicans who ultimately feel the most heat.

The counter-intuitive dynamic is rooted in the midterms map: Whereas many of the key Senate battlegrounds are deeply Republican states where President Donald Trump won handily in 2016, the fight for the House is taking place in suburban districts in states like California, Pennsylvania and New York — areas with high proportions of college-educated women who could be particularly moved by Ford's allegations and her emotional testimony to the Senate.

"This hearing, and the entire Supreme Court fight, actually may do more to hurt the GOP in House ... than it does in the Senate, despite the fact that senators are voting on it," said one House Democratic campaign veteran. "Mitch McConnell and (Senate Judiciary Committee Chairman) Chuck Grassley may be helping end the Republican majority in Congress."

While some Republicans said a capable House candidate would come out of this debate unscathed, other party operatives fear the spectacle that engulfed the Senate Judiciary Committee Thursday could deepen misgivings among voters -- particularly women -- about the Republican Party, which has consistently underperformed in generic polling leading during this midterm election cycle.

A GOP campaign operative put it bluntly: "The women are gone. It's all about the base."

And that sentiment was backed up by the judge's combative performance, which at times mimicked the President himself with direct attacks against the media, Democrats on the committee and even a passing reference to Bill and Hillary Clinton.

"Voters certainly vote on the candidates in individual races, but they also vote as a choice between two parties," the Democratic campaign veteran added. "And the brand of the Republican Party is now the party of this hearing and the party of Brett Kavanaugh."

A host of Democratic candidates, particularly those running to represent heavily suburban districts, jumped on the Kavanaugh hearing, particularly Ford's compelling an emotional testimony.

In the race against Republican Rep. Peter Roskam in the Chicago suburbs, Democrat Sean Casten has been honing in on Kavanaugh since his nomination in July, initially focusing on concerns that Kavanaugh might vote to overturn Roe v. Wade. Decades ago, Roskam's district was represented by Henry Hyde, the namesake of the Hyde Amendment restricting federal funding for abortions. But the district's complexion has shifted since Hyde's time, and Casten has sought to capitalize on a growing pro-choice contingent.

Casten has also used Kavanaugh's nomination to criticize Roskam as a rubber stamp for the President; this week, the Casten campaign has promoted an ad on Facebook calling on Roskam to withdraw his support for the embattled nominee.

But the dramatic hearing Thursday, including Ford's tearful testimony, has upped the ante further.

"I think it means we're going to keep the pressure on (Roskam), that people should believe women," said Greg Bales, Casten's campaign manager. "She (Ford) showed incredible courage, and Peter Roskam doesn't have the courage to stand up to Trump."

"This is an important issue," Bales added, "and we're going to continue bringing attention to it."

Roskam's campaign did not respond to a request for comment.

Casten's campaign isn't alone. In New Jersey's 7th District, another crucial battleground, Democrat Tom Malinowski's campaign posted a video on Facebook of his Republican opponent, Rep. Leonard Lance, praising Kavanaugh and saying "I tend not to believe the charges" of sexual misconduct against him.

"Leonard Lance's first instinct — the woman is lying," reads a message at the conclusion of the video, which has so far been viewed around 8,000 times. "That's the real Leonard Lance."

And in Colorado's 6th District, where Democrat Jason Crow is challenging Republican Rep. Mike Coffman, Crow released a statement praising Ford's testimony as "incredibly brave" and calling on Kavanaugh's nomination to be withdrawn.

Meanwhile, as the hearing played out Thursday, the Democratic Congressional Campaign Committee sent a fundraising email signed by House Minority Leader Nancy Pelosi, in which she invoked the spectacle and Republicans' continued support for Kavanaugh.

"We can — and WILL — punish these despicable Republicans and win a Democratic Congress that'll render them powerless," Pelosi wrote, before soliciting donations to the committee.

Republican campaign operatives reacted to the hearing with a mixture of concern and resolve. Ford "certainly came across as an accomplished professional and someone who isn't fabricating her story," acknowledged one party operative working on the midterms. "But there are a ton of holes there."

And some Republicans downplayed the possible political impact of the hearings, predicting that Kavanaugh's passionate defense of his character could enliven GOP voters -- a crucial page in the GOP playbook this year.

"There's an argument it will hurt among college-educated women," said Michael Steel, a Republican strategist and former senior aide to Speaker John Boehner. "There's another argument that any college-educated woman who's turned off by this probably was turned off by the President's statements and actions a long time ago." Meanwhile, Steel added, "this could serve to energize and rally the core Trump supporters to get out and vote."

Suburban women were overwhelmingly opposed to Kavanaugh even before the dramatic testimonies Thursday -- with a recent Fox News poll showing just 32% of suburban women in favor, and 63% against confirming him.

As Kavanaugh's confirmation has become a political flashpoint, much of the focus politically has been on vulnerable Senate Democrats, who will vote yea or nay on his nomination. Three of Democratic senators from red states supported Justice Neil Gorsuch last year.

But with Ford's allegations and her testimony on live television Thursday, along with Kavanaugh's defense, the story is now captivating a nation -- and extending beyond the Senate battlegrounds. In addition to House candidates, Democratic candidates for governor also weighed in on Kavanaugh and Ford, including Andrew Gillum in Florida.

"It's a major moment -- a major television moment, and a major political moment," said Doug Thornell, a Democratic strategist and former DCCC spokesperson. "... It's not something that is limited to the Senate, it sort of transcends that."

A top Republican working in one swing district admitted that it is hard to know "whether or not something like this affects a race."

"You have to know your candidate is someone who has built a strong reputation in his district, is respected, well liked and thoughtful," they said. "When you have a candidate like that, they end up running on their record, and people tend to drown out the surrounding noise that isn't directly related to your race."

They added: "Ultimately, good candidates get people to vote for them as opposed to go against someone else."

In light of the hearings Thursday, however, Steel predicted that some Republican House candidates in suburban districts might begin to distance themselves from Kavanaugh, including by "suggest(ing) that the process could have been carried out differently, and obviously ultimate blame starts with Senate Democrats who were aware of these allegations for weeks."

Steel added, "I would imagine there would be a lot of House members who say that they want a conservative justice, but maybe Kavanaugh wasn't the right choice for this time."