It is not possible to make published public ideas private or vice versa.
If users were able to make specific public ideas private, there would be a way to hide (formerly public) losing trades from the community. And if users were able to make specific private ideas public, there would be a way to showcase winning (formerly private) trades, while keeping losing ideas private. As a consequence, the ideas tab on the profile page of a user would not necessarily present a complete picture, which would violate the principle of transparency that TradingView upholds.
Please note it is of course possible to republish a private idea as a public idea.