McCaughey's comments came after Sanders told CNN's Martin Savidge that "emotions were still high" after President-elect Donald Trump's victory.
"I think on both sides, it was a really nasty general election campaign. We were having conversations, and still are, about white supremacy, and racism," said Sanders, who supported Democratic-nominee Hillary Clinton after the Vermont senator failed to secure the party's nomination.
Critics have accused the Republican victor of rooting his campaign in racist rhetoric and discriminatory policy proposals -- a charge the Trump campaign has denied.
A furious McCaughey, who supports Trump, disagreed with Sanders.
"People like Symone are still making false accusations of racism," McCaughey said, before accusing Democrats of "behaving like racists themselves" in their critique of Sen. Jeff Sessions, Trump's pick for attorney general.
"He is a civil rights hero," McCaughey said. "And yet right away, they try today to accuse him of being a racist. I even hear some people say that a white person should not lead the Democratic Party. What could be more racist than that? One reason Donald Trump won is that Americans are sick and tired of reverse racism. And that's what they heard from Hillary Clinton."
Sanders responded: "Do you know what, Betsy -- I'm sorry. We've got to cut this out." People of color in the US are "victims of systemic racism," she added.
"There are still some real underlying issues. ... These are things people in America are still dealing with. That's why we're having conversations about racism and white supremacy. And the Trump campaign and the Republicans have to at least own that, so we can move forward" Sanders said.
McCaughey replied: "We don't own it because those accusations of white racism are false."
Savidge interjected: "Is there a way, Symone, Betsy, is there a way you can listen to a counterargument and not shout back?"
"I did listen to it," McCaughey said. "But I don't appreciate the accusations of racism because they are untrue, and the only reason that people like Symone constantly label their opponents as racists is they don't have an argument they can win on the merits."
Sanders responded: "Betsy just 'these peopled' me."