"We have a lot of issues with Iran," Tillerson said on CBS's "Face the Nation." "They're a yard long. The nuclear issue is one foot of that yard. We have two feet of other issues that we must deal with. And it has to do with Iran's destabilizing activities."
For his part, Supreme Leader Ali Khamenei, the ultimate power in Tehran's theocracy, took to his English-language Twitter account to label Washington as, in turn, domineering, bullying, oppressive, hounding and cruel — and corrupt and lying to boot.
"Every day US govt. exposes a new side of its viciousness & proves Imam Khomeini's words true: U.S. govt. is the great Satan," he tweeted.
The criticisms were lobbed at a critical moment for the Iran deal, which eased economic sanctions in return for Iran agreeing to restrictions on its nuclear program.
It is being kept alive for the time being, after President Trump put aside his disdain for the deal on Thursday and waived U.S. sanctions that were suspended under the agreement and must be revisited every 120 days. If he hadn't, the United States would have been in breach of its promises.
But the administration is still reviewing its policy toward Iran and the nuclear deal, and Trump has said he is inclined to say next month that Iran is not complying with its commitments. If he does, Congress will have 60 days to decide whether to reimpose sanctions, in effect breaking the U.S. commitment.
The International Atomic Energy Agency has said eight times that Iran is complying with the deal, officially known as the Joint Comprehensive Plan of Action, as Tillerson allowed on Sunday.
The administration contends that Iran is violating the "spirit" of the deal, because in its preface it is stated that the nations negotiating it "anticipate that full implementation of this JCPOA will positively contribute to regional and international peace and security."
Tillerson said that sentence explains, in a nutshell, why sanctions were lifted.
"But since the nuclear deal has been concluded, what we have witnessed is Iran has stepped up its destabilizing activities in Yemen, it stepped up its destabilizing activities in Syria, and exports arms to Hezbollah and other terrorist groups, and it continues to conduct a very active ballistic missile program," Tillerson said. "None of that, I believe, is consistent with that preamble commitment."
Supporters of the deal say it was never intended to solve every issue between the United States and Iran. The diplomats who negotiated said at the time that the deal was narrowly focused on Iran's nuclear program because it was considered better to confront Tehran without the possibility of nuclear weapons.
Iran, which has always denied seeking to build nuclear weapons, has complained it has not received the economic benefits it expected from the deal because the United States has not done enough to convince the business community that it will remain in effect so long as Iran keeps its promises.
Iran's sense that it was shortchanged in the deal was behind a series of tweets by Khamenei on Sunday, in between congratulating graduating police cadets and criticizing Aung San Suu Kyi's silence on the plight of Burma's Rohingya.
Calling the U.S. approach to the nuclear deal "totally oppressive hounding & cruel" in one tweet, Khamenei in another tweet accused "corrupt, lying U.S. officials" of hypocrisy.
"Enemies must know if bullying works elsewhere in world, it won't work for Iran," he said in another tweet. "Retreat has no place when it comes to our national interests."
And he suggested that any move to decertify Iran's compliance or withdraw from the deal will not go unanswered.
"The Iranian nation stands strong."