Kavanaugh meets with GOP leaders as swing-vote senators, under pressure, promise 'due diligence'

President Trump's Supreme Court nominee Brett Kavanaugh met with Republican leaders on Capitol Hill Tuesday, as critical swing-vote senators promised to withhold judgment on his nomination until they could vet his lengthy record.

Fox News has learned Vice President Mike Pence will meet tonight with Senate Judiciary Committee Chairman Chuck Grassley, R-Iowa, and Ranking Member Dianne Feinstein, D-Calif., to discuss Kavanaugh's nomination. Pence is set to huddle with other key senators on the nomination in the coming days.

The initial consultations on Capitol Hill came as progressive, pro-choice activists ratcheted up their rhetoric, saving the conservative justice's appointment could directly cause women to die.

One top Democrat, Sen. Richard Blumenthal of Connecticut, even called Kavanaugh "the worst nightmare" of Florida high school students who survived a mass shooting this year. Kavanaugh has a largely pro-gun record; he voted in the landmark Heller case to overturn a D.C. ordinance that placed significant restrictions on the ownership of semi-automatic weapons. The Supreme Court eventually took the case and struck down the ordinance.

Kavanaugh, 53, on Tuesday first huddled with Senate Majority Leader Mitch McConnell of Kentucky. Joining him were Pence and former Arizona Sen. Jon Kyl, who's set to serve as his "sherpa" by guiding him through the trappings of the confirmation process.

"It’s okay to take it slow."

— Alaska Republican Sen. Lisa Murkowski

Kavanaugh also met for roughly 30 minutes with Grassley.

WATCH: PROTESTERS SPAR AMID HEAVY POLICE PRESENCE AT SUPREME COURT

While Republicans have set a goal of confirming Kavanaugh by this fall, Grassley said speed isn't the goal. The vetting process, he said, is "going to be thorough and going to be done right." He did not offer a timeline for confirmation hearings, although the White House has said it expected Kavanaugh to be on the Supreme Court by October.

McConnell called Kavanaugh "one of the most thoughtful jurists" in the country and blasted Democrats as "eager to try and turn judicial confirmations into something like political elections." The GOP leader warned against engaging in "cheap political fear-mongering."

For his part, Pence called Kavanaugh, a Catholic man who had clerked for retiring Associate Justice Anthony Kennedy, a "good man."

Republicans hold a slim 51-49 Senate majority, but they hope to gain support for Kavanaugh from a handful of Democrats who are up for re-election in states where Trump is popular. Several of those Democrats, including Sens. Joe Manchin of West Virginia, Joe Donnelly of Indiana and Heidi Heitkamp of North Dakota, voted for Neil Gorsuch's nomination last year.

Liberal activists have sought defections by two moderate Republican senators who support abortion rights: Maine's Susan Collins and Alaska's Lisa Murkowski.

Murkowski, who voted to confirm Kavanaugh to a seat on the influential D.C. Circuit Court of Appeals in 2006, said Tuesday that his nomination to the Supreme Court "does put it into a different category," adding that "there’s more to review."

"It’s OK to take it slow," Murkowski said, criticizing those who have quickly rushed to support or deride the nominee.

"The fact that people are already saying no or yes -- we’re not even 24 hours in," she added. "I'm going to take my time."

READ MORE: WHY KAVANAUGH HAS ALARMED BOTH PRO-LIFE AND PRO-CHOICE ADVOCATES

Meanwhile, Collins said she would conduct a "careful, thorough vetting" of Kavanaugh. But she also sounded a positive note, saying he has "impressive credentials and extensive experience, having served more than a decade on the D.C. Circuit Court of Appeals."

Collins has said she would oppose any nominee she believed would overturn Roe v. Wade. It was clear that liberal activists had senators like her in mind Monday night, as progressive groups raced to send out doomsday predictions as soon as Kavanaugh's nomination was announced.

In a statement, the Women's March said ominously: "Trump’s announcement today is a death sentence for thousands of women in the United States." The organization's statement at one point incorrectly referred to Kavanaugh as "XX," indicating the document had been written in advance and hastily sent out.

Fox News' Chad Pergram and The Associated Press contributed to this report.