The nuclear deal “is in the background of all this for the Europeans and for Rouhani, too, especially given sanctions renewal in mid-January,” Ms. Geranmayeh said.
So far, the European Union and its member states have issued statements condemning the violence and warning the Iranian government not to overreact. But the statements do not support the protesters or condemn the Islamic government as explicitly as Mr. Trump’s Twitter messages do.
European officials say they are for now resisting American efforts to issue a more critical joint statement at the United Nations or at the U.N. Human Rights Council in Geneva, as called for by Nikki Haley, Mr. Trump’s ambassador at the United Nations.
Mr. Trump has already “decertified” Iran last October for not, in his view, living up to its commitments under the nuclear deal, negotiated in 2015 under President Obama. But even then he signed a waiver on the sanctions, which he must do every 120 days.
On Tuesday, the White House press secretary, Sarah Sanders, said Mr. Trump had not decided what he will do about the sanctions.
“We certainly keep our options open in terms of sanctions,’’ she said. “In terms of signing a waiver later in January, the President hasn’t made a final decision on that. He’s going to keep every option on the table.”
If he does not sign, renewed sanctions will affect both American and European companies, given the power of the dollar and the American banking system.
The reimposition of sanctions could also further undermine the standing of Mr. Rouhani, while allowing Iran’s hard-liners to argue that it had been wrong to trust the United States on the nuclear deal, which could founder.
The Europeans have been pushing Washington hard, especially in Congress, to preserve the nuclear deal with Iran despite Mr. Trump’s regular and longstanding criticism of it as “the worst deal ever.”
With protests against the Islamic government in Iran and its economic performance, including the vast sums it spends supporting Shia allies abroad like Syria, Hezbollah and the Yemeni Houthis, Mr. Trump may decide that to waive the sanctions again would only give more economic help to a regime he disparages.
“People on the ground in Iran are still trying to figure out where all this is going,” said Ms. Geranmayeh, the Iran expert.
“The E.U. is quite different from the Trump message, expressing concern over what’s happening but trying not to harm those demonstrating in Iran, while creating a certain degree of expectation — ‘Rouhani, we are watching you,’ ” she added.
The Europeans are important for Mr. Rouhani, too. They are the biggest potential investors in Iran and also have been fierce defenders of the nuclear deal that Mr. Rouhani supported against deep skepticism from the hard-liners and Iran’s Supreme Leader Ayatollah Ali Khamenei.
The European Union’s statement, issued Tuesday night by the high representative for foreign affairs, Federica Mogherini, says that the deaths of demonstrators are “inexcusable” and calls on the Tehran government for restraint, while noting that Mr. Rouhani has said Iranians have the right to peaceful political protest.
It says that such rights are fundamental and “apply to every country, and Iran is no exception.”
“We expect all concerned to refrain from violence and the right of expression to be guaranteed, also in light of the statements made by the Iranian Government,” the statement said.
Ms. Mogherini has been criticized for delay in making the statement, but the European Union works by consensus, and diplomats have been caught on holidays.
The British Foreign Secretary, Boris Johnson, was criticized for responding too blandly to the situation, but on Monday he also called for freedom of expression and the right to demonstrate peacefully to be respected, and said that the issues raised by the protesters were legitimate.
In a telephone call with Mr. Rouhani on Tuesday, French President Emmanuel Macron also asked for restraint and agreed to postpone a visit scheduled for this week by the French foreign minister, Jean-Yves Le Drian.
On Wednesday morning, the German government issued a similar statement, urging the Iranian government to initiate talks with the protesters and respond proportionately.
“The federal government considers it legitimate when people courageously protest their economic and political woes on the streets as is happening in Iran at the moment and they have our respect,” said a government spokeswoman, Ulrike Demmer.
The European Union and other signatory members, France, Britain and Germany, have regularly defended the Iran nuclear deal against American criticism and said Iran was in full compliance.
They have argued that the deal is only about the nuclear program, not other Iranian actions, some of which are already sanctioned. But Mr. Trump thinks the accord was too limited and did not do enough to constrain Iranian actions in other areas.
“If the Iranian regime would use more violence to crack down, then I’m sure the European Union would take a tougher stance,” said Raz Zimmt, a research fellow at the Institute for National Security Studies and the Alliance Center for Iranian Studies at Tel Aviv University.
“But for now the Iranian regime doesn’t want to use too much force,” fearing escalation inside the country and a harsher reaction outside, he said.
But he expressed skepticism that the European efforts to push Mr. Rouhani would significantly hasten economic reforms and civil liberties.
“I doubt that Rouhani will be able to do something significant without the approval of the supreme leader, and the supreme leader, as we all know, is quite concerned with any change in Iran,” Mr. Zimmt said.
The European Union has also been firmly opposed to Mr. Trump’s decision to recognize Jerusalem as Israel’s capital. European officials have been troubled also by a symbolic, nonbinding resolution by the Likud Party, led by Israeli Prime Minister Benjamin Netanyahu, to annex Israeli settlements in the West Bank, across the green line that marks Israel’s pre-1967 boundaries.