It would seem to me that this order tramples on the First Amendment. No matter how much the executive wants to trample on twitter, they are still a private company and can publish or not publish what they wish.
I don't see anything in the order that prohibits speech, instead focusing on questioning if Twitter et al. qualify for platform protections. Plus some other ad-money ammunition.
An argument could be made that this EO itself violates the first amendment as it's clearly in response to the tweet fact checks, thereby silencing twitter?
I'm going to say I believe the courts will block this, but it will look good to Trump's base in an election year.