Tillerson's remarks followed harsh criticism of Trump from National Economic Council Director Gary Cohn, who said in an interview on Saturday that he nearly quit over the President's handling of the events in Charlottesville.
"Citizens standing up for equality and freedom can never be equated with white supremacists, neo-Nazis, and the KKK," Cohn said in the Financial Times interview.
Cohn, who is Jewish, said that the Administration "must do better in consistently and unequivocally condemning these groups".
Trump has condemned hate groups in the wake of Charlottesville but twice added equivocation about blame for violence and once said some "fine people" were among the white-supremacist marchers.
At a campaign rally before a largely white crowd last week in Phoenix, Trump decried the removal of Confederate statues - the flash point in Charlottesville - and blamed the news media for "trying to take away our history and our heritage".
Tillerson had told a group of State Department interns on August 19 that "we do not honour, nor do we promote or accept, hate speech in any form".
Tillerson did not invoke Trump or levy direct criticism then, but his discussion of "hate speech" just days after the Charlottesville rally made his meaning clear.
"Those who embrace it poison our public discourse, and they damage the very country that they claim to love," Tillerson had said.
Today, Tillerson rejected criticism from a United Nations committee last week that the Trump Administration had failed in its response to Charlottesville and set a poor example for the rest of the world.
"We express America's values from the State Department - our commitment to freedom, our commitment to equal treatment of people the world over," Tillerson said in the Fox interview, "and that message has never changed".
The leaders of Britain and Germany had previously said in response to Charlottesville that violence and bigotry must always be condemned.
Defence Secretary Jim Mattis expressed implicit criticism of Trump last week in impromptu remarks to US troops captured on video.
Mattis says the country has "problems," and asks those in uniform to "hold the line until our country gets back to understanding and respecting each other and showing it."
Mattis has not commented on the video, which was apparently recorded as he addressed troops in Jordan.
Treasury Secretary Steven Mnuchin has defended Trump's remarks, but most other top aides have not directly addressed the controversy.