"It has invaded Ukraine, including the taking of Crimea, and supported Syrian forces that brutally violate the laws of war. Our NATO allies are right to be alarmed at a resurgent Russia," Tillerson said in the prepared statement.
Tillerson said the absence of American leadership had opened the door to this behavior and sent unintentional signals.
President-elect Donald Trump has faced criticism over flattering remarks he has made about Russia and Putin, and for his denial until recently that Russia was behind cyberattacks on the Democratic National Committee and other U.S. targets.
The U.S. intelligence community concluded the Russian state acted to disrupt the U.S. democratic process during the 2016 election, in part to damage Democrat Hillary Clinton's chances of being elected.
Rubio, a 2016 presidential contender who frequently sparred with Trump, asked Tillerson whether he would advise the president-elect to sign a bill that imposed mandatory visa bans and asset freeze sanctions on actors who engage in cyberattacks on U.S. infrastructure and democratic process, regardless of what country they hail from.
Tillerson said it is important to deal with those attacks on a country-by-country basis, taking into consideration all other elements of the relationship with the nation, including trade and national security ties.
Rubio said he found that answer "troubling" because it implies he may advise the president not to impose sanctions on a country with which the administration was trying to improve relations.
Trump has frequently said he will seek to improve relations with Russia.
Asked whether Trump agrees with his views on Russia's involvement in Ukraine and in Syria's civil war, Tillerson said he and the president-elect "have not had the opportunity to discuss this specific issue or the specific area."
Democratic Sen. Bob Menendez said he would have thought Russia would have been at the top of their conversation considering recent developments.
"Has that not happened?" Menendez asked.
"That has not occurred yet," Tilleson responded.
Tillerson added that to his knowledge, Exxon Mobil never lobbied against sanctions on Russia.
Menendez, a long-time member of the Foreign Relations committee, said Wednesday that Exxon had in fact lobbied "aggressively" against Russian sanctions after its invasion of Ukraine and last year against related legislation, employing lobbyists to support these efforts.