Donald Jr. also has a big role in the family business, as an executive vice president whose portfolio includes expanding the company’s real estate, retail, hotel and golf interests.

He has also had a taste of reality stardom, appearing as an adviser on “The Celebrity Apprentice,” an iteration of his father’s hit show “The Apprentice.” He lives with his wife and five children in New York, where he has flirted with the notion of running for mayor. “I like to keep optionality,” he said when asked on CNN if he would consider a run.

The Trump campaign has called the scrutiny of his Twitter posts unfair.

“We’re truly living in remarkable times,” said Jason Miller, a spokesman for the campaign. “The media’s run out of things to attack Mr. Trump on, and so now they scour the social media accounts of his family looking for things to blow out of proportion.”

Appearing this month on “Good Morning America,” Donald Jr. was asked by George Stephanopoulos about his use of the frog picture, which Mr. Stephanopoulos called a “well-known symbol of the white supremacist movement.”

“I’ve never even heard of Pepe the Frog,” Mr. Trump said. “I mean, bet you 90 percent of your viewers have never heard of Pepe the Frog.” He added: “I thought it was a frog in a wig. I thought it was funny.”

Mr. Trump’s father has also employed his Twitter feed in ways many have found offensive. He has reposted messages from white supremacists’ accounts, and in July he posted an image of Mrs. Clinton, a pile of cash and a six-pointed star in the shape of the Star of David. He quickly deleted the post and said it was not intended to be anti-Semitic, but he later said it should not have been deleted. Around that time, his son was retweeting anti-Semitic social media users.

In March, Donald Jr. appeared on a radio show where one of the interviewers had had David Duke, the former Ku Klux Klan leader now running for the Senate in Louisiana, as a guest on his affiliated show. The interviewer, James Edwards, is the host of “The Political Cesspool,” which has been described by the Southern Poverty Law Center as “racist and anti-Semitic.”