Senator Rand Paul of Kentucky took a firm position Monday on the side of people who say vaccination is a personal choice, overshadowing similar comments made earlier in the day by Gov. Chris Christie of New Jersey, a fellow Republican, and inserting himself into a hot debate that will likely resurface if the senator runs for president.

Mr. Paul, who is an eye doctor, said on CNBC that “the state doesn’t own your children” and that “profound medical disorders” sometimes occur after vaccines.

He also appeared to lose his temper later in the interview, telling the host Kelly Evans to “calm down” so that he could finish his response.

“You’ve taken the interview and you’ve made it into something where you got no useful information because you were argumentative,” Mr. Paul said.