Thatcher reportedly referred to French-African tennis star Jo-Wilfried Tsonga as a "golliwog frog" and "half golliwog" in the green room of The One Show earlier this month.
O' Grady told The Sun: "I've met Carol and she's lovely, she really is. I don't believe she's a racist.
"But she shouldn't be saying things like that. She needs a good slap on the wrist."
He added that Prince Harry should not have made allegedly racist comments to his colleagues while serving in the army.
O'Grady said: "For God's sake, what's up with these people? There are certain words you don't say - in public or private.
"Re-educate yourself. There's no excuse. You don't think like that, never mind talk like that. Their friends might say they are OK with it, but they shouldn't be."