The Brazilian used a derogatory term in an interview that was given in November last year when speaking about the 2021 British GP.
He also used a homophobic phrase towards Hamilton when describing how Rosberg beat him to the championship in 2016.
Piquet, whose daughter Kelly is in a relationship with Red Bull’s Max Verstappen, slammed Hamilton’s driving at Silverstone where he made contact with the Dutchman on an explosive first lap.
Verstappen was sent hurtling in the barriers and taken to hospital for checks while Hamilton went on to win his home race for an eighth time.
Piquet claims Hamilton was at fault and referred to him with the offensive term, which was initially missed by local Brazilian media.
The 68-year-old apologised for what he says was an “ill thought out” comment, made in November last year.
A statement from Piquet said: “What I said was ill thought out, and I make no defence for it.”
He added: “I strongly condemn any suggestion that the word was used by me with the aim of belittling a driver because of his skin colour.
“I apologise wholeheartedly to anyone that was affected, including Lewis, who is an incredible driver, but the translation in some media that is now circulating on social media is not correct.
“Discrimination has no place in F1 or society and I am happy to clarify my thoughts in that respect.”
What was Lewis Hamilton’s response?
Hamilton who arrived at Silverstone in a colourful outfit today, said: “I am incredibly grateful to all those who have been supportive within the sport, particularly the drivers.
“It has been two years since we took the knee in Austria. Of course we are still faced with the challenges.
“I have been on the receiving end on racism and that negative and archaic narratives and undertones of discrimination.
“I am not sure why we are continuing to give these older people a platform.
“They are speaking on the sport but we are looking to go in a different direction. It’s the bigger picture.
“It is now a knee-jerk reaction from companies all around the world.
“They probably all already have a script ready for that crisis management. It is not enough. It’s now about action.
“We need more than ever to be bringing people together. It is not helpful the comments we are seeing from these people.
“People who are not relevant in the sport for years are trying to bring me down but I am still here.”