Tillerson spoke to reporters after a United Nations meeting of countries involved in the Joint Comprehensive Plan of Action, the agreement that curbed Iran's nuclear program in exchange for sanctions relief, and one President Donald Trump has excoriated.
The 72nd annual General Assembly meeting at the United Nations was dominated by concerns about North Korea's nuclear program and the fate of the Iran nuclear deal, which Trump declared "an embarrassment to the United States" and has indicated that he wants to abandon.
The "Iran deal was one of the worst and most one-sided transactions the United States has ever entered into," Trump said in his first address to the UN.
On Wednesday, Trump told reporters at the UN that he had made a decision about how to proceed when he next has to certify in October that Iran is in compliance.
Tillerson said Trump has approached the decision of whether to stay in the Iran nuclear deal in the "right way," and has been "very deliberative," listening to a variety of viewpoints.
"I think the time has come for a decision to be made," Tillerson added. Trump "has not shared that decision (about the future of the deal) with anyone externally," including British Prime Minister Theresa May, who asked, Tillerson said.
The top US diplomat said his first opportunity to shake hands with Iranian Foreign Minister Mohammad Javad Zarif left him with the impression of a "well-educated, very sophisticated" man.
He said that at the meeting there was "a good opportunity to meet, shake hands" and "the tone was very matter-of-fact, there was no yelling." He added with a chuckle, "we didn't throw shoes at one another."
"Whether we can agree, that's another thing," Tillerson said.
The leaders gathered at the JCPOA meeting had an "open and candid exchange," Tillerson said. It was "not a technical discussion," he said, "it was a political discussion."
He reiterated the Trump administration's concerns about the deal and Iran's behavior writ large, and said the administration is planning further steps to address that behavior without falling out of compliance with the deal.
While everyone around the table agreed Iran is in technical compliance with the deal, Tillerson said the US believes Iran is not in compliance with the spirit or expectations of the deal.
"I found it useful to hear their perspectives," he said of his fellow members, "and I hope they found it useful as well."
He added that US allies are looking "more carefully and seriously" at ways to push back against Iranian aggression.
"We also have been making the case to our allies that, 'You share our concerns about Iran's behavior. Please join us in sending that strong message to them. You're not violating the agreement,'" Tillerson said. "We believe they're looking more carefully and seriously at that as a means to push back on this other behavior of Iran's that they agree with us is unacceptable."
Speaking after the meeting, the European Union's foreign policy chief, Federica Mogherini, made clear that European parties to the deal aren't interested in revisiting the agreement, which took two years to finalize.
"There is no need to reopen the agreement because it's fully delivering" and Iran is abiding by it, Mogherini said. "This is an agreement that prevented a nuclear program and potentially prevented a military intervention," Mogherini said. "Let's not forget that."
She said that, "it's not for one party or the other to certify this." That's a job for the International Atomic Energy Agency, "with its technical independent role" to verify that Iran is sticking to the agreement.
And referring to the brewing nuclear crisis with North Korea, Mogherini said, "we already have one potential nuclear crisis, we definitely do not need to go into a second one."
Tillerson was asked what message North Korea should draw from the US attempt to renegotiate an agreement that has already been settled. Tillerson is conducting what he calls a "peaceful pressure" campaign to try to force Pyongyang into negotiations.
"We would take a very different view of the nature of the North Korea agreement," Tillerson said, pointing out that there are few similarities between the two countries. "The threats the same, but the nature of the agreement is going to be quite different."
Tillerson was asked later whether he could have a similar discussion with a North Korean representative.
"I don't know whether I could have the same kind of matter of fact discussion with North Korea because we don't know how their means of communication and behavior will be."
Tillerson said there also is evidence of fuel shortages in North Korea following the passage of new UN Security Council sanctions.
"We have some indications that there are beginning to be evidence of fuel shortages," he said. "We knew these sanctions were going to take some time to be felt." And looking forward, he said, "I think we'll see latent evidence of the impact of other sanctions in place."