"If you violate international agreements, if you fail to live up to commitments, if you become a threat to others, at some point a response is likely to be undertaken," Tillerson said on Sunday AP reported.
There was little doubt the missile strikes would be seen in Pyongyang as a message. The North has long claimed that the US is preparing some kind of assault against it and justifies its nuclear weapons as defensive in nature.
US Navy ships are a common presence in the Korean Peninsula region and serve in part as a show of force. On Saturday night, the Pentagon said a Navy carrier strike group was moving toward the western Pacific Ocean to provide more of a physical presence in the region.
North Korea has pledged to bolster its defenses to protect against such airstrikes. The North called the US action in Syria "absolutely unpardonable" and said it proves that its nuclear weapons are justified to protect the country against Washington's "evermore reckless moves for a war."
The comments were made by a Foreign Ministry official and carried by North Korea's state-run Korean Central News Agency on Sunday. The report did not name the official, which is common in KCNA reports.
Tillerson said advances in North Korea's ballistic missile program concerned the US the most. Asked if development of an intercontinental missile was a "red line" for Trump, Tillerson said: "If we judge that they have perfected that type of delivery system, then that becomes a very serious stage of their further development."
Trump and South Korea's leader, acting President Hwang Kyo-Ahn, spoke by phone on Friday, according to the White House, which said they agreed to stay in close contact about North Korea and other issues.
Trump has said China, with its ties to North Korea, should deal with its neighbor, but that the US was prepared to act on its own if necessary.