But his willingness to stand behind the president, coupled with the assault on his beliefs by Democratic candidates, has only further highlighted a key truth about Mr. Pence: He is less a good cop than he is a good soldier.

“It’s possibly the case that just because the show has become so mesmerizing, there hasn’t been enough attention for there to be scrutiny on him and what he represents,” Mr. Buttigieg said. “But there’s a nontrivial chance of Pence becoming the president, any day, and we should understand what we’re getting there.”

Mr. Buttigieg said that Mr. Pence was a politician “whose views are very much out of the mainstream.” He ticked off issues like gay marriage (Mr. Pence has said he is opposed to it); climate change (Mr. Pence has said the causes of it remain to be seen); and evolution (Mr. Pence has said God created heaven and earth) as policy issues on which the vice president needed to clarify his positions.

Mr. Pence declined to comment publicly, but a senior administration official familiar with his thinking said he does not plan to take the bait. “He just lets the criticism roll off,” said that official, who spoke on the condition of anonymity so as to not disclose private conversations.

But Mr. Biden, who is seen in the White House as Mr. Trump’s most formidable potential 2020 challenger, earned a public rebuke from the vice president.

“Well, look, they’ve got an awfully big field,” Mr. Pence said on Thursday in an appearance on “Fox & Friends,” adding that “the way Joe Biden caved in to liberal activists after making a kind comment for me is evidence of the pressure that all of their candidates are going to face.”

Still, some observers of the race say the scrutiny of Mr. Pence may continue.

“I’m surprised he hasn’t been in the middle of it more,” said Tim Miller, a longtime Republican strategist and a “Never Trumper.” “He’s skillfully avoided, to a certain degree, the amount of association he should have with Trump.”