After a bumpy few days that began with Mr. Paul, a physician turned politician, appearing to question the safety of vaccines and then snapping at a television interviewer who pushed him to clarify, one of his biggest liabilities was suddenly impossible to ignore: Does someone who can be so impetuous and unapologetic have the finesse and discipline to win over people who are more naturally inclined to vote for someone else?

In a series of interviews over the last several weeks, including one in which he invited a New York Times reporter to watch him get a Hepatitis A booster shot to dispel the notion that he was anti-vaccine, Mr. Paul displayed little interest in embracing the please-all-people approach to politics.

“Everybody is going to be a critic about something,” he said. “I don’t wear the right clothes; my hair’s not great. You are who you are.”

“You can spend your whole life worrying about too many little middling things,” he went on.

Questions about his style and temperament are not incidental to doubts about whether he can build the diverse political coalition he seeks. He has constantly fought to ease concerns from old-guard Republicans that he will work against the party, much as his father’s supporters did when they seized control of many state Republican organizations in recent years.

“I’m impressed with the guy,” said Richard Hohlt, a prominent Republican donor. “But the problem is, can he be like Reagan and say to the Libertarian Party: ‘Cool it. I’m going to run to win, and then I’ll come back to you and we’ll work together. Don’t trash me in the process because you know where I stand, and I’m the one guy you can trust.’ ”