In addition, US President Donald Trump affirmed on Thursday that Iran "is not living up to the spirit" of the nuclear agreement, which he described as terrible and badly negotiated.
In the first reaction to Tillerson's remarks from a senior Iranian official, Iranian Foreign Minister Mohammad Javad Zarif, who tweeted that the United States should "fulfill its own commitments".
"An unchecked Iran has the potential to travel the same path as North Korea - and to take the world along with it", Tillerson said.
"Worn-out USA accusations can't mask its admission of Iran's compliance" with a 2015 nuclear deal, Zarif wrote on Twitter.
Tillerson's statement went on to describe some of those provocations, including support for terrorism, interference in Iraq, backing the Assad regime in Syria, support for the Houthis in Yemen, harassment of US navy ships in the Persian Gulf, and cyber-attacks against USA interests and allies.
"They are not living up to the spirit of the agreement, I can tell you that", Trump said of the Iranians on Thursday, though he did not mention any specific violations. However, the USA admitted that Tehran was complying with the 2015 agreement, the BBC said.
On the climate agreement, the White House postponed a meeting Tuesday where top aides were to have hashed out differences on what to do about the non-binding worldwide deal forged in Paris in December 2015.
Tillerson's letter follows an assessment by the International Atomic Energy Agency in February that Iran has so far complied with restrictions on its uranium enrichment and is not pursuing construction on its heavy-water research reactor.
Fabio Borini hints Sunderland locker room problems source of relegation woes
Or will West Ham add to the horror season for Sunderland? Subs (not used): Adrian (gk), Cresswell, Feghouli, Fletcher, Rice . Harry Maguire eventually took advantage of some poor Stoke defending at a corner to level matters in the 51st minute.
Iran helped to create and continues to fund Hezbollah, the Lebanese military and political organisation which the United States has listed as a terrorist organisation.
Both Iran and Hezbollah are now fighting in support of Syrian President Bashar al Assad's forces. But he said its "nuclear ambitions" remained "a grave risk to worldwide peace and security".
As he often had during the president campaign, Trump ripped into the deal struck by Iran, the US and other world powers in 2015 and said "it shouldn't have been signed".
Trump has panned what he has called "the awful Iran deal" reached by the United States and five other world powers under his predecessor, former usa president Barack Obama.
But Rex Tillerson has come pretty close to saying the agreement is not worth keeping, even though he's had to admit it's working.
Opponents of the agreement have called for renegotiating the accord with the goal of making permanent its 15-year moratorium on uranium enrichment close to the level needed to make a bomb, but reimposing sanctions that were explicitly tied to Iran's nuclear program - as Tillerson suggested in his announcement - would face particular opposition from the US' European allies and give the government in Tehran grounds to walk away from the accord.
Tillerson said the Trump administration is conducting a "comprehensive review" of the agreement.