The seven-time world champion was speaking ahead of this weekend’s British Grand Prix - days after 69-year-old Piquet’s offensive reference to Hamilton emerged from a podcast recorded last November.
Triple world champion Stewart, 83, said recently that Hamilton should “resign” after struggling for results this season, while Ecclestone, 91, told ITV’s Good Morning Britain on Thursday that he would “take a bullet” for Vladimir Putin, also describing the Russian president as a “first-class person”.
“I don’t know why we are continuing to give these older voices a platform,” said Hamilton.
“They are speaking for our sport, but we are looking to go somewhere different and they are not representative of who we are now in the sport and where we are planning to go.
“If we are looking to grow our audiences in places like the US and South Africa, we need to be giving the younger people a platform that is more representative of today’s time and who we are trying to be.
“It is not just about one individual, or the use of that term (by Piquet), but the bigger picture.”
Hamilton, 37, continued: “I put to F1 and to the media that we should not be giving these people a platform - these old voices, who, whether it is subconscious or conscious, do not agree people like me should be in this sport.
“I don’t think in the last couple of weeks a day has gone by where some of the older people, who have not been relevant in our sport for decades, are trying to say negative things and bring me down.
“But I am still here, still standing strong and trying to do my work and pushing diversity. Discrimination exists. It is still all around us. Enough is enough.
“I have tried to take that high road and avoid it, but no one should have to brush off racism and it shouldn’t be for me to have to brush it off. We need big organisations to take a stand.”
Hamilton also took aim at Good Morning Britain for inviting Ecclestone on to their programme.
He continued: “There needs to be some accountability. You know what you are going to get with that, and I don’t know what their (GMB’s) goal is and if they were seeking to create and divide here in the UK.
“We don’t need any more of it. To hear from someone that believes in the war, and the displacement of millions of people and killing of thousands people, and supporting that person (Putin) who is doing that, is beyond me. I cannot believe I heard that today.
“It is affecting all those people out there and people around the world. This is going to put us back decades, and we have yet to see the real brunt of the pain.
“So, why? We do not need to be supporting that but looking into the future. We need more positivity. If you don’t have anything positive to contribute to where we want to be going and where we want to be, don’t give them any space.”
Ecclestone ruled F1 for more than four decades, but the sport moved to distance itself from his remarks.
An F1 spokesperson said: “The comments made by Bernie Ecclestone are his personal views and are in very stark contrast to the position of the modern values of our sport.”