Washington (CNN) Sen. Kamala Harris on Wednesday defended her calls for Twitter to suspend President Donald Trump's account, a plan she reiterated on stage at the fourth Democratic primary presidential debate hosted by CNN and The New York Times.

Harris first made her call following the President's tweets about the whistleblower complaint at the center of the Ukraine scandal. During Tuesday night's debate, Harris called out Sen. Elizabeth Warren of Massachusetts, one of the field's front-runners, for not following her lead in saying Trump's Twitter account should be taken down. Warren brushed off Harris' proposal, saying she didn't just want to "push Trump off Twitter" but also out of the White House.

The contentious moment led to some criticism for Harris, including from former US National Security Council spokesperson Tommy Vietor, who called Harris' push during the debate "small ball."

Harris, however, told CNN's John Berman on "New Day" Wednesday that Trump's "threatening remarks" about the whistleblower and others is a "serious matter."

"It's a serious matter. And it is a matter of the safety of those individuals," she said. "You can look at the fact that the shooter in El Paso was influenced by the words that the President of the United States, unfiltered, uses through this medium on Twitter. He has 65 million followers. And we have to take seriously the implications that are about the threatening witnesses, intimidating witnesses and obstructing justice."

She added: "When witnesses are threatened by the President of the United States, it is a very serious matter. And people will be influenced by those threats. And we have to say that it is not OK and that he has revoked his privilege to use this platform because it is a privilege that can be and should be taken away."

Harris also said companies like Facebook and Twitter need to adopt the same set of standards to govern what is allowed on their platforms.

"And the point also has to be, you can't have one set of standards for Facebook and another for Twitter," she said on Wednesday. "All of these social media companies and these online platforms, which are so powerful in their ability to impact perception about an issue and to influence behaviors. Let's be clear about that. There have to be standards, and the standards have to be the same. You can't have one standard for Facebook and another for Twitter."

While companies like YouTube, Facebook and Twitter collaborate closely and are closely aligned on policing content from terrorist groups, each is a separate company and has their own set of standards for political speech.

In May, for instance, when a fake video of House Speaker Nancy Pelosi circulated, the three major platforms acted differently. YouTube removed the video, Facebook let the video stay online but downranked it meaning it would be seen by less people and Twitter allowed the video to stay online.

Earlier this month, Harris sent a letter to the Twitter CEO Jack Dorsey listing six tweets by the President that she says violated Twitter's rules. The White House called Harris' demand "authoritarian" and defended Trump's use of Twitter.

"It is not surprising that Kamala Harris, someone who believes in bigger government and more regulation, would like to silence her political opponents," the White House said in a statement. "In fact, it's rather authoritarian of her. President Trump's use of technology to communicate directly with the American people and share his Administration's unprecedented accomplishments should be praised, not criticized."

On Tuesday, Twitter issued a clarification in a blog post on rules for tweets by world leaders, saying "that the accounts of world leaders are not above our policies entirely." The social media network reiterated its announcement from June that some tweets from world leaders that break its rules would not be removed from the platform if the company decided they were in the "public interest." Twitter said it would instead put a disclaimer on the tweets explaining that although they broke the company's rules they would not be removed.

Twitter has not labeled any of Trump's tweets in this way.