Sen. Elizabeth Warren became the punchline for more “Pocahontas” jokes Tuesday after her high-profile DNA rollout backfired, angering Native Americans and rekindling questions about her claims to Cherokee ancestry.

President Trump called on her to apologize for “perpetrating this fraud against the American Public” after her DNA test showed she has between 1/64th and 1/1,024th Native blood, or 0.09 to 1.5 percent, about the same as the average U.S. white person.

“Now that her claims of being of Indian heritage have turned out to be a scam and a lie, Elizabeth Warren should apologize for perpetrating this fraud against the American Public,” he tweeted.

Boston radio host Howie Carr said that far from proving her ancestry, the test actually showed that she isn’t Native American. The average European American has 0.18 percent Native ancestry, according to a 2014 study.

“The whole thing is just ridiculous,” Mr. Carr said on his show. “Again, she’s just got to count on the alt-left media, the credulous Boston Globes of the world and CNNs, promoting this as a real example that she’s a Native American when the reality is it proves that she is not a Native American.”

Even CNN had its doubts. The network featured an article by Kate Maltby called “Elizabeth Warren’s big mistake,” while anchor Chris Cillizza said she “might have actually made things worse with her DNA gambit.”

Others had fun with the latest chapter in Ms. Warren’s Cherokee saga. Sen. Orrin Hatch, Utah Republican, tweeted that he had taken a DNA test showing that he was “1/1032 T-Rex/The rest: other dinosaurs.”

These DNA tests are quite something. pic.twitter.com/tCHmW7pJbj — Senator Hatch Office (@senorrinhatch) October 15, 2018

The deluge of Warren-themed memes on social media included numerous photos of white Jeep Grand Cherokee SUVs whose owners have named them “Elizabeth Warren.” A Twitter account emerged called “Things Less White Than Elizabeth Warren,” which includes examples such as The Brady Bunch.

Sen. Lindsey Graham, South Carolina Republican, said he would take a DNA test to see if he were more Native American than Ms. Warren.

“I’m going to take a DNA test,” he told “Fox & Friends. “I’ve been told my grandmother was part Cherokee Indian. It may be just talk. But you’re gonna find out in a couple of weeks.”

Democrats, meanwhile, were irked over her decision to signal her 2020 presidential intentions just three weeks before the critical Nov. 6 midterms, faulting her for shifting the focus away from the candidates while providing political cannon fodder for Republicans.

Former Obama campaign manager Jim Messina said Tuesday that Ms. Warren is taking Democrats “off our message” and diverting media attention from the issues.

“I understand what she’s doing as a political move. If I were running her campaign, I would probably do this,” said Mr. Messina on MSNBC. “She wants this taken off the table. I just wouldn’t do this now. Not 21 days before the election.”

Jim Messina Criticizes Warren For Taking Democrats Off Message pic.twitter.com/hdOION7LQ0 — Alex Wilkes (@AlexandraCSmith) October 16, 2018

Others have argued that Ms. Warren’s decision to confront the “Pocahontas” issue may wind up benefiting her by showing that she’s not afraid to take on Mr. Trump.

“She’s showing that she can play offense as well as defense, and she won’t be bullied by the president and that she’ll also taunt him with these results,” said Tufts University political science professor Jeff Berry on WBUR-FM in Boston.

“What’s important here is that she’s saying when he punches, I’m going to counterpunch,” he said.

Nobody was more critical of Ms. Warren’s decision to trumpet her DNA results than the Cherokee Nation, which made it clear that genetic testing does not prove ancestry under tribal law and accused her of “undermining tribal interests with her continued claims of tribal heritage.”

Ms. Warren began listing herself as Native American as a law professor at the University of Pennsylvania, and was later hired to teach at Harvard Law School. Both the senator and Harvard officials have insisted that her Native heritage claim played no part in her hiring, which her critics have disputed.

“While #ElizabethWarren no longer identifies herself as Native, she still publicly claims her family is ‘part Native American,’” tweeted activist Rebecca Nagle, a Cherokee Nation member. “There is nothing innocent about a White woman claiming her family experienced genocide and ethnic cleansing—when they did not.”

Cornell Law School professor William A. Jacobson called the Cherokee Nation condemnation “a severe blow to Warren’s credibility.”

“Warren ignored the first law of holes,” said Mr. Jacobson, who runs the right-tilting Legal Insurrection blog. “She was in a hole over her frivolous claim to be Native American, but she just kept digging herself deeper with the dubious DNA publicity rollout.”

The brouhaha comes with a recent CNN poll showing Ms. Warren placing fourth in a hypothetical 2020 Democratic presidential sweepstakes, trailing former Vice President Joseph R. Biden, Sen. Bernie Sanders, and Sen. Kamala Harris.

After she released the DNA results, she called out the president, challenging him to make good on his July promise to donate $1 million to charity if she proved the test “shows she’s an Indian.” Mr. Trump countered that she has not done that.

Sen. Chris Coons, Delaware Democrat, commended Ms. Warren for attempting to resolve the issue.

“She’s put to rest his I think juvenile, offensive and inappropriate suggestions using his nickname for her that somehow she didn’t know her own family history,” said Mr. Coons on CNN. “I think it’s best for everybody if we just move on here and recognize that this is just another episode that was beneath the president.”

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