DES MOINES, Iowa—Presidential candidate Donald Trump refused Sunday to apologize for criticizing Sen. John McCain’s war record, a politically risky stance in keeping with his long record of standing behind provocative remarks and brushing off guidance from aides.

Mr. Trump operates as his own chief strategist and political consultant—the “chief theoretician of Trumpism,” according to one person who works with him. He’s not known to submit himself to policy briefings or do the homework typical of successful politicians, according to another person who has worked for Mr. Trump on previous political efforts.

The result is a Republican presidential campaign of freewheeling, off-the-cuff commentary that always carries the potential to run off the rails.

On Sunday, Mr. Trump didn’t back down from remarks he made a day earlier at the Family Leadership Summit in Ames, Iowa, where he said that Mr. McCain, who spent five years as a prisoner of war in Vietnam, was “not a war hero.” He added that Mr. McCain was considered “a war hero because he was captured. I like people who weren’t captured.”

Mr. Trump, who also said Sunday he was ignoring calls to drop out of the presidential race, has been fuming about Mr. McCain since the senator referred to the crowd at Mr. Trump’s rally in Phoenix last week as “crazies,” which Mr. Trump took as a personal insult.