The former F1 driver was fined R$5million – then the equivalent of £778,846 – by a Brazilian court in March.
His attempts to appeal the verdict have now failed, Globo reports.
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Charges have been brought against Piquet by four human rights groups.
Piquet commented on Hamilton in an interview conducted in November 2021.
He later apologized to Hamilton.
Piquet, 70, is a three-time F1 champion.
His daughter, Kelly Piquet, is the companion of Max Verstappen.
Hamilton said when the original decision was made earlier this year: “At the time it happened I commented on it – I still believe we shouldn’t generally give people which are just full of hating a platform.
“I would like to thank the Brazilian government. I think it’s quite amazing what they’ve done in holding someone accountable, showing people that it’s not tolerated.
“Racism and homophobia are not acceptable and there is no place for it in our society.”