The politics of medicine, morality and free will have collided in an emotional debate over vaccines and the government’s place in requiring them, posing a challenge for Republicans who find themselves in the familiar but uncomfortable position of reconciling modern science with the skepticism of their core conservative voters.

As the latest measles outbreak raises alarm, and parents who have decided not to vaccinate their children face growing pressure to do so, the national debate is forcing the Republican Party’s 2016 presidential hopefuls to confront questions about whether it is in the public’s interest to allow parents to decide for themselves.

Gov. Chris Christie’s trade mission to London was suddenly overshadowed on Monday after he was quoted as saying that parents “need to have some measure of choice” about vaccinating their children against measles. The New Jersey governor, who is trying to establish his credibility among conservatives as he weighs a run for the Republican nomination in 2016, later tried to temper his response. His office released a statement clarifying that “with a disease like measles there is no question kids should be vaccinated.”

Senator Rand Paul of Kentucky, a physician, was less equivocal, telling the conservative radio host Laura Ingraham on Monday that parents should absolutely have a say in whether to vaccinate their children for measles.

“While I think it’s a good idea to take the vaccine, I think that’s a personal decision for individuals,” he said, recalling his irritation at doctors who tried to press him to vaccinate his own children. He eventually did, he said, but spaced out the vaccinations over a period of time.

The vaccination controversy is a twist on an old problem for the Republican Party: how to approach matters that have largely been settled among scientists but are not widely accepted by conservatives.

It is a dance Republican candidates often do when they hedge their answers about whether evolution should be taught in schools. It is what makes the fight over global warming such a liability for their party, and what led last year to a widely criticized response to the Ebola scare.

As concern spread about an Ebola outbreak in the United States, physicians criticized Republican lawmakers — including Mr. Christie — who called for strict quarantines of people who may have been exposed to the virus. In some cases, Republicans proposed banning people who had been to the hardest-hit West African countries from entering the United States, even though public health officials warned that would only make it more difficult to stop Ebola’s spread.

On climate change, the party has struggled with how to position itself, with some Republicans inviting mockery for questioning the established science that human activity is contributing to rising temperatures and sea levels.

There is evidence that vaccinations have become more of a political issue in recent years. Pew Research Center polls show that in 2009, 71 percent of both Republicans and Democrats favored requiring the vaccination of children. Five years later, Democratic support had grown to 76 percent, but Republican support had fallen to 65 percent.

The debate does not break entirely along right-left lines. The movement to forgo vaccinations has been popular in more liberal and affluent communities where some parents are worried that vaccines cause autism or other disorders among children.

President Obama acknowledged the concern as a candidate in 2008. “Some people are suspicious that it’s connected to the vaccines, this person included,” he said, pointing to a member of the crowd, and he did later go on to say that the issue warranted further study. But asked about immunization over the weekend in an interview with the “Today Show” on NBC, Mr. Obama urged parents to “get your kids vaccinated.”

Hillary Rodham Clinton also weighed in with a jab at vaccine naysayers, writing Monday night on Twitter, “The science is clear: The earth is round, the sky is blue, and #vaccineswork.”

Howard Dean, a presidential candidate in 2004 and a former chairman of the Democratic National Committee, said there are three groups of people who object to required vaccines: “One is people who are very much scared about their kids getting autism, which is an idea that has been completely discredited. Two, is entitled people who don’t want to put any poison in their kids and view this as poison, which is ignorance more than anything else. And three, people who are antigovernment in any way.”

“But the truth,” added Mr. Dean, a physician, “is you can be conservative without putting kids in harm’s way.”

The issue has more political potency among conservative voters who are highly skeptical of anything required by the government.

The vaccine question surfaced in the 2012 Republican primary when rivals of Rick Perry, then the Texas governor, pounced on him for issuing an executive order requiring sixth-grade girls to be vaccinated against the human papillomavirus — the first regulation of its kind in the country. One of his opponents, Michele Bachmann, a congresswoman in Minnesota, went as far as saying the vaccine could cause “mental retardation,” a claim with no scientific merit. But in a sign of the issue’s political weight, Mr. Perry apologized for the mandate.

Asked about the measles vaccine controversy on Monday, a spokesman for Mr. Perry affirmed his commitment to “protecting life” and pointed to efforts by his administration to increase immunization rates.

Mike Huckabee, the former Arkansas governor who is considering a run for president, has noted that the link between autism and vaccines was discredited. As governor, he received his flu shot at the State Capitol and encouraged all Arkansans to get vaccinated.

But for Republicans like Mr. Paul who appeal to the kind of libertarian conservatives who are influential in states like Iowa and New Hampshire, which hold the first two contests in the battle for the nomination, there is an appeal in framing the issue as one of individual liberty.

Asked about immunizations again later on Monday, Mr. Paul was even more insistent, saying it was a question of “freedom.” He grew irritated with a CNBC host who pressed him and snapped: “The state doesn’t own your children. Parents own the children.”