In a major foreign policy speech on US relations with Europe under Trump's "America First" foreign policy, Tillerson pledged that sanctions against Russia will remain in place until it helps restore peace in Ukraine.
In his speech, Tillerson described Russia as one of the "major challenges" facing Europe and the US.
Europe and the United States seek a normalised relationship with Russia, he said.
However, Russia, Tillerson said, has shown it seeks to define a new post-Soviet global balance of power, one in which Russia, by virtue of its nuclear arsenal, seeks to impose its will on others.
Russia does so by force or by partnering with regimes who show a disregard for their own citizens, as is the case with Bashar al-Assad's continuous use of chemical weapons against his own people, he alleged.
The dissolution of the Soviet Union liberalised Russian society and created new trade opportunities that benefit Russians, Europeans, and Americans, the top US diplomat.
"But Russia has often employed malicious tactics against the US and Europe to drive us apart, weaken our confidence, and undermine the political and economic successes that we have achieved together since the end of the Cold War," Tillerson said.
"Playing politics with energy supplies, launching cyber- attacks and disinformation campaigns to undermine free elections, and serially harassing and intimidating diplomats are not the behaviours of a responsible nation," he said.
Tillerson went further: Attacking a neighbouring country and threatening others does nothing to improve the lives of Russians or enhance Russia's standing in the world.
"We want Russia to be a constructive neighbour of Europe and of the larger transatlantic community."
He said the ball was in Russia's court and it could choose whether to "continue to isolate and impoverish itself by sowing disorder abroad and impeding liberty at home, or become a force that will advance the freedom of Russians and the stability of Eurasia."
Following President Donald Trump's recent decision regarding US policy toward Iran, there is actually much more that binds the United States and Europe together than drives them apart, the Secretary of State said.
"We ask our European partners to join us in standing up to all of Iran's malign behaviour. The Iranian regime is antithetical to Western principles in its totalitarian suppression of individual, political, and religious freedom," he said.
"Neither the US nor Europe wants another type of North Korea nuclear threat on its hands, nor are any of our nations at ease with Iran's attempts at hegemony in the Middle East through support for terrorist organisations, militias on the ground in Iraq and Syria, and an active ballistic missile development program," he added.
Responding to questions, Tillerson said there are areas of mutual cooperation with respect to Russia.
"We're working hard in Syria to defeat ISIS and we are on the cusp of having ISIS once and for all defeated in Syria... We are working together with Russia on how to prevent the civil war from re-erupting in Syria...," he said.
"We think there are areas of greater cooperation on counterterrorism with Russia. There may be opportunities for cooperation in Afghanistan. We've not yet come to what that might be, but we're talking about it," Tillerson said.