But Mr. Trump’s invocation of Mr. Obama and the nuclear deal muddies his message, analysts said, by turning the spotlight away from the Iranian government’s economic failures — which have given rise to this powerful, if inchoate, protest movement — to the lingering debate in Washington over the nuclear agreement.

Mr. Trump never fully resolved that debate himself. In October, he refused to certify the deal, but he left it to Congress to legislate changes to it. Lawmakers have made little progress and European leaders have refused to revisit it. Between Jan. 11 and 17, Mr. Trump faces new deadlines on whether to recertify the deal and to continue to waive sanctions.

“He was going to be put on the spot, anyway, explaining why he was keeping the deal alive without these improvements,” said Philip H. Gordon, a senior National Security Council official in the Obama administration. “If the Iranians are killing people in the streets when it comes time for Trump to extend the sanctions waivers, it is hard to see him doing it.”

Yet killing the deal, Mr. Gordon said, could enable the Iranian government to galvanize domestic support against the United States rather than face questions about why it has not been able to improve Iran’s economy. “Right now, they cannot blame us or the international sanctions,” he said. “This could allow them to make the U.S. the enemy.”

Even critics of the deal said they worried that the protests would tempt Mr. Trump to abandon it rather than try to improve it.

“If there is a bipartisan bill that is ready for congressional action, that would go a long way toward persuading the president to issue the waivers,” said Mark Dubowitz, the chief executive of the Foundation for Defense of Democracies. “If there’s not, what’s happening in Iran will give the president all the more reason to say, ‘I’ve had it with this deal.’”

On Capitol Hill, however, many lawmakers expect Mr. Trump will reluctantly agree to waive sanctions, leaving the deal in place. In theory, that would allow the Senate additional time to try to meet Mr. Trump’s request that the legislature make alterations to the agreement.