WASHINGTON — 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.