The executive order is not touching the protections of the first amendment. The courts are still to protect them for that as always.
The order regards the additional protections of section 230 which even protect twitter for content that is not protected by the first amendment. Trump is essentially trying to say if Twitter takes sides by fact-checking some tweets, then they are also responsible for all the other "facts" they allow to be posted on their platform without fact-checking. And by the way the courts are still perfectly capable of deciding in favor of twitter regarding blame for all those other posts too. Twitter just won't be shielded by a special law from such decisions.
The order regards the additional protections of section 230 which even protect twitter for content that is not protected by the first amendment. Trump is essentially trying to say if Twitter takes sides by fact-checking some tweets, then they are also responsible for all the other "facts" they allow to be posted on their platform without fact-checking. And by the way the courts are still perfectly capable of deciding in favor of twitter regarding blame for all those other posts too. Twitter just won't be shielded by a special law from such decisions.