Yet other Democrats are quick to note that party candidates who are running more cautious and centrist campaigns in conservative states are faring no better than the liberal Mr. O’Rourke. In Tennessee former Gov. Phil Bredesen, the Democratic nominee for the Senate, came out in support of Justice Kavanaugh but has seen his standing in the polls plummet.

And, these Democrats say, Washington-based strategists are in denial about both why Mr. O’Rourke has raised so much money and what it says about the party’s fund-raising culture. Mr. O’Rourke has harnessed social media to viral effect, live-streaming much of his waking life, from rallies to burger runs.

“People will, because of the internet and because of how people now access information, donate to any candidates and causes that inspire them most,” said Jon Favreau, a former speechwriter for Barack Obama who has been an outspoken supporter of Mr. O’Rourke. “The idea that there is some entity in D.C. that can direct the fund-raising for how people give is thinking that is so 10, 20 years ago.”

The deeper tension, which Democrats will mostly only discuss privately while Republicans say it publicly, owes to the question of Mr. O’Rourke’s ultimate ambitions: Does he believe that an unapologetically liberal campaign in a traditionally red state is the best way to win in 2018? Or might he achieve political martyrdom at the hands of a senator Democrats hate and then seek a bigger platform?

“Clearly he’s not running for the United States Senate from Texas,” Senator John Cornyn of Texas said Friday on Laura Ingraham’s talk radio show, adding: “My only suspicion, Laura, is he’s running for the Democratic nomination for president in 2020.”