“What we’re seeing here is the standard process of Trump launching a series of lies into our media ecosystem and then watching it bounce around and amplify through the system,” said Siva Vaidhyanathan, a professor of media studies at The University of Virginia. “So to respond with anything reasonable to the fact that the president basically made stuff up is to extend the message and to throw a bomb” into the political debate.

The president’s tweet about Ms. Waters contained a false statement, an insult and what sounds like a threat. But each time it was shared — even, and perhaps especially, by critics who wanted to vent their anger — the message was amplified and spread.

What Ms. Waters Said

Here is how this particular episode began: some of Mr. Trump’s advisers have recently been hassled at restaurants or at a movie theater by protesters opposed to his administration’s policy of separating migrant children from their parents at the border.

At a rally on Saturday in Los Angeles, Ms. Waters praised those protesters and encouraged them to do more, using physical terms like “push back.”