It is the fight President Trump had been waiting for, a showdown with a top political rival in a deep blue state over an issue core to his political agenda.

In bypassing the authority of Gov. Gavin Newsom of California, a Democrat, to call in the National Guard to quell protests in the Los Angeles area over his administration’s efforts to deport more migrants, Mr. Trump is now pushing the boundaries of presidential authority and stoking criticism that he is inflaming the situation for political gain.

Local and state authorities had not sought help in dealing with the scattered protests that erupted after an immigration raid on Friday in the garment district. But Mr. Trump and his top aides leaned into the confrontation with California leaders on Sunday, portraying the demonstrations as an existential threat to the country — setting in motion an aggressive federal response that in turn sparked new protests across the city.

As more demonstrators took to the streets, the president wrote on social media that Los Angeles was being “invaded and occupied” by “violent, insurrectionist mobs,” and directed three of his top cabinet officials to take any actions necessary to “liberate Los Angeles from the Migrant Invasion.”

“Nobody’s going to spit on our police officers. Nobody’s going to spit on our military,” Mr. Trump told reporters as he headed to Camp David on Sunday, although it was unclear whether any such incidents had occurred. “That happens, they get hit very hard.”

The president declined to say whether he planned to invoke the 1807 Insurrection Act, which allows for the use of federal troops on domestic soil to quell a rebellion. But either way, he added, “we’re going to have troops everywhere.”