Let's reread the first amendment again:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
What is the first word? CONGRESS. The first amendment only looks at the ability of lawmakers. But it can't stop private people from excluding you! Twitter can do what it wants and the first amendment does not apply. The Government can't compel Twitter to host speech either, because of Miami Herald Publishing Co. v. Tornillo 418 US 241 (1974) banning compelled speech.
To the Popehat!
This question is core Ken "Popehat" White material. He is an avid first amendment attorney and should have something to say about this... And actually, he did already talk about pretty much the very topic, but not in regards to Mr. Trump. In his "Make No Law" podcast Episode 11: Deplatformed he starts with the basic idea of if people can be thrown from a website and starts out pretty clearly before dissecting the problem and explaining why:
You don't have a protected legal right to be on someone's Website.
Often cited that one would have a right to be on a website is Marsh v Alabama. However, that doesn't apply because Twitter is not a company town or taking duties that are usually part of the government! So the holding that Twitter might be a state actor by doing what the state does and has to abide by the constitution simply does not apply. As Ken White says (emphasis mine):
Instead, the Supreme Court has repeatedly ruled that a private entity only engages in State Action and therefore is only subject to the First Amendment when it performs a traditional exclusive public function.
Twitter is NOT doing a traditional exclusive public function. In fact, 2019 had the SCOTUS verdict in Manhatten Community Access Corporation v. Halleck 139 S.Ct. 1921 (2019). And Ken White quotes from the Verdict (emphasis mine):
Justice Brett Kavanaugh: Providing some kind of forum for speech is not an activity that only governmental entities have traditionally performed; therefore, a private entity who provides a forum for speech is not transformed by that fact alone into a state actor. After all, private property owners and private lessees often open their property for speech. Grocery stores put up community bulletin boards, comedy clubs host open mic nights.
And interviewing Eric Goldman, author of Technology and Marketing Law and professor at Santa Clara University School of Law, he gets this gold evaluation of the case:
[...] in terms of the basic principle that private publishers aren’t state actors, it’s possible that we could read it as a 9-0 opinion.
Then he discusses PruneYard... and why it fails.
Now, like Marsh, [Prune Yard]'s been limited to its own facts and on several occasions the California Supreme Court has come within one vote of overturning it entirely. There is absolutely no indication that it can be extended to social media sites.
He also discusses how section 230 doesn't give rise to any claim, or rather, why it makes Twitter impervious: the law expressly allows Twitter to ban any stuff! And it is actually very misunderstood by people, as Ken White explains in a separate article.
Here is the second crucial part of Section 230 Subsection C2. It protects websites and users when they moderate content. Here’s the language. ‘No provider or user of an interactive computer service shall be held liable on account of any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected.’ What does that mean?
It means what it says, that a website can delete your post or ban you if it thinks you’ve posted something lewd, harassing or objectionable and you can’t sue them for it. Once again, this is essential to the operation of the Internet.
Section 230 was not passed to require sites to be neutral. To the contrary, it was passed to give websites freedom and moderating vigorously under the theory that it was better than government intervention. And yet, this idea that Section 230 requires sites to moderate in a neutral fashion has become very popular even though it has no basis in reality or law.
Or, maybe you would like the more concise form from his separate article summarizing Section 230 (emphasis mine):
Very simply put, Section 230 is the law that says that, if I post something defamatory on Twitter, the victim can sue me, but not Twitter. It also says, again put simply, that Twitter has the right to moderate stuff on its site as it sees fit. The language of Section 230 is fairly clear, for something written by Congress. There is very little controversy amongst actual courts about what it means. The legal impact of Section 230 has been well-established by courts for decades, and efforts to evade it have been consistently rejected.
Or the rather clear language of the statute itself (emphasis mine):
No provider or user of an interactive computer service shall be held liable on account of—
(A)any action voluntarily taken in good faith to restrict access to or availability of material that the provider or user considers to be obscene, lewd, lascivious, filthy, excessively violent, harassing, or otherwise objectionable, whether or not such material is constitutionally protected; or
(B)any action taken to enable or make available to information content providers or others the technical means to restrict access to material described in paragraph (1).
Yes, reread that: [Section 230] was passed to give websites freedom and moderating vigorously under the theory that it was better than government intervention. This means: Twitter can throw you or anyone else out because of section 230 c. And NOBODY has a recourse. Not you, not Mr. Trump, not even the government! And yes, Eric Goldman agrees in that (emphasis mine):
Eric Goldman: It’s censorship to tell people what they can’t publish and it’s censorship to tell people what they must publish. Those are two different sides of the same coin; they’re both ways of curbing editorial freedom. So, anything that curbs editorial freedom in my mind is censorship trying to force services to be neutral which means to subvert their editorial judgment is a form of censorship.
And censorship? THAT is banned under the first amendment. Censorship is making a law that abridges the freedom of speech - and THAT is unconstitutional.
Oh, and the 14th Amendment? NOT a help either. It is only about state action, not private parties. Civil Rights Act? No protection against someone for their political beliefs or use of their speech.
Final Nail: Twitter TOS
Finally Twitter has every right to throw you or anyone out, including and because of a contract that was agreed to by the user, which expressly includes a termination clause (emphasis mine), the TOS:
We may suspend or terminate your account or cease providing you with all or part of the Services at any time for any or no reason, including, but not limited to, if we reasonably believe: (i) you have violated these Terms or the Twitter Rules and Policies or Periscope Community Guidelines, (ii) you create risk or possible legal exposure for us; (iii) your account should be removed due to unlawful conduct, (iv) your account should be removed due to prolonged inactivity; or (v) our provision of the Services to you is no longer commercially viable. We will make reasonable efforts to notify you by the email address associated with your account or the next time you attempt to access your account, depending on the circumstances. In all such cases, the Terms shall terminate, including, without limitation, your license to use the Services, except that the following sections shall continue to apply: 2, 3, 5, and 6. If you believe your account was terminated in error you can file an appeal following the steps found in our Help Center (https://help.twitter.com/forms/general?subtopic=suspended). For the avoidance of doubt, these Terms survive the deactivation or termination of your account.
Do you spot the most powerful word choice there? for any or no reason (as long as it is not a protected characteristic). They don't even need a reason to kick you. They just like to have one, but they have under the contract the Twitter user agreed to the right to throw him off the platform without recourse because an admin had a bad day and wants to kick a random user.
Protected characteristic however does only work to make a ban illegal if it is the own: Saying "I am a green Martian" can't be the reason for ban if you are one, but banning for "Burn all green Martians" would not be a violation of the discrimination ban - and political speech is not a protected characteristic.