Under the agreement, the United States can still penalize Iran for behavior such as its development of ballistic missiles or support for terrorism, but it cannot simply reapply the same sanctions that were lifted under a different guise. Iran has the right to appeal to a joint committee and make the argument that the United States is in violation.

European officials have long argued that the agreement was intended only to restrict Iran’s nuclear program, not the panoply of other issues. If the international community were to do more to confront Tehran over its efforts to destabilize neighbors in the Middle East, European officials have said, it would be better to face an Iran without nuclear weapons. They have shown little enthusiasm for revisiting the deal, much less undercutting it.

That Mr. Trump’s actions will satisfy conservatives who have been urging him to rip up the Iran deal seemed unlikely. In a column in The Hill, a Capitol Hill newspaper, John R. Bolton, a former ambassador to the United Nations who has interviewed with the president for several jobs in the administration, argued that the nuclear agreement “remains palpably harmful to American national interests.” Withdrawing from it, he said, “should be the highest priority.”

Israel and its supporters in Washington have also bristled at a new cease-fire in southwest Syria that was brokered by Mr. Trump and President Vladimir V. Putin of Russia, fearing it will leave Iran as a major player on the ground in the six-year civil war. Mr. Spicer said the administration would address that with Israel. “There’s a shared interest that we have with Israel, making sure that Iran does not gain a foothold, military base-wise, in southern Syria,” Mr. Spicer said.

Tehran’s clerical government argues that Mr. Trump has already violated the nuclear agreement by pressuring businesses not to engage with Iran even though the nuclear sanctions have been lifted. “That is violation of not the spirit but of the letter of the J.C.P.O.A. of the nuclear deal,” Mohammad Javad Zarif, Iran’s foreign minister, said on CNN’s “Fareed Zakaria GPS” on Sunday, using the initials for the agreement.

Mr. Zarif, visiting New York, told a gathering of the Council on Foreign Relations on Monday that he has yet to talk with Mr. Tillerson, unlike his frequent conversations with former Secretary of State John Kerry, with whom he negotiated the nuclear accord.