That's wrong. They have the right to fact check. They simply might not have section 230 immunity with respect to the fact check because that is self-generated content.
They still enjoy immunity with respect to user-posted content, because that is part of the explicit language of the statute and a Presidential Order can't override statutory language.
They still enjoy immunity with respect to user-posted content, because that is part of the explicit language of the statute and a Presidential Order can't override statutory language.