“I guarantee you everybody that ever bought a Harley-Davidson voted for Trump,” he said. “And they’re very unhappy about it.”

A demand for more trade concessions was one part of a campaign message Mr. Trump appeared to outline for Republicans in November’s elections, which also included putting another conservative on the Supreme Court and endorsing the administration’s crackdown on immigration.

That plan appears to include seizing as a political weapon the growing call among liberal activists to dissolve ICE, shifting the battle to Democrats’ fault lines on immigration rather than the rifts among Republicans that have been exposed by Mr. Trump’s policies.

But even as Mr. Trump encouraged the liberal embrace of abolishing ICE, he had promised earlier in the weekend on Twitter that there was “zero chance, it will never happen!”

Democrats have been united against Mr. Trump’s family separation policy, which he officially ended through an executive order. On Saturday, demonstrators gathered across the nation — including outside the White House and miles from the Bedminster resort where he spent the weekend — to protest the zero-tolerance policy and the government’s struggle to reunite families.

But only a small number of Democratic lawmakers have called for abolishing ICE. Still, their ranks have been growing, especially since the No. 4 House Democrat was unseated in a primary last week by a liberal challenger who had called for eliminating ICE.

The agency, which has struggled to balance its role in transnational investigations and deportations, is best known for its division responsible for arresting, detaining and deporting unauthorized immigrants. Under the Trump administration, ICE has faced growing backlash over its tactics, including the arrest of undocumented immigrants as they drop their children off at school, and detaining and deporting those with minor offenses.