Public/private space is an unclear term of art which is not relevant to US law. There are four relevant categories created by the interaction between the private / government distinction, and "open to the public" vs. "not open to the public" (for example: government parks, military bases, private shopping malls, your own home). The First Amendment says
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.
Obviously the government cannot prohibit you from criticizing or praising the president, per se, and as applied to things in the government park category, that means that you can give a soapbox speech on whatever topic you want in the park. On the other hand, the First Amendment does not prohibit the government from protecting individual property rights, which means that you cannot invade my home and use it as a venue for a protest, without my permission: that is, the First Amendment does not nullify trespassing laws and constitutional property rights. The courts have long held that the government can designate properties such as military bases as closed areas where demonstrations can be banned, even (US v. Apel) when an easement grants access, so the right to protest on government land is not absolute.
One relevant decision about so-called public spaces on private property and the First Amendment is Lloyd v. Tanner 407 U.S. 551. That court held that
There has been no dedication of petitioner's privately owned and
operated shopping center to public use so as to entitle respondents to
exercise First Amendment rights therein that are unrelated to the
center's operations, and petitioner's property did not lose its
private character and its right to protection under the Fourteenth
Amendment merely because the public is generally invited to use it for
the purpose of doing business with petitioner's tenants. The facts in
this case are significantly different from those in Marsh, supra,
which involved a company town with "all the attributes" of a
municipality, and Logan Valley, supra, which involved labor picketing
designed to convey a message to patrons of a particular store, so
located in the center of a large private enclave as to preclude other
reasonable access to store patrons. Under the circumstances present in
this case, where the handbilling was unrelated to any activity within
the center and where respondents had adequate alternative means of
communication, the courts below erred in holding those decisions
controlling.
Subsequently and similarly we find Hudgens v. NLRB, 424 U.S. 507, a shopping mall protest case, where it was ruled that the First Amendment does not negate private property rights. The court observed that
It is, of course, a commonplace that the constitutional guarantee of
free speech is a guarantee only against abridgment by government,
federal or state...Thus, while statutory or common law may in some
situations extend protection or provide redress against a private
corporation or person who seeks to abridge the free expression of
others, no such protection or redress is provided by the Constitution
itself.
The court did refer to prior Marsh v. Alabama (First Amendment argument upheld regarding speech on privatly owned property), 326 U.S. 501, where the location in question was a "company town", indistinguishable from a regular town except that the property was owned by a private corporation rather than a municipal corporation. Citing Food v. Logan, 391 U.S. 308, the Hudgens court observed that
streets, sidewalks, parks, and other similar public places are so
historically associated with the exercise of First Amendment rights
that access to them for the purpose of exercising such rights cannot
constitutionally be denied broadly and absolutely.
The question which the court then raised was
under what circumstances can private property be treated as though it
were public? The answer that Marsh gives is when that property has
taken on all the attributes of a town, i.e., "residential buildings,
streets, a system of sewers, a sewage disposal plant and a ‘business
block' on which business places are situated."
and there is
nothing in Marsh which indicates that, if one of these features is
present, e.g., a business district, this is sufficient for the Court
to confiscate a part of an owner's private property and give its use
to people who want to picket on it.
The court reminds us that
it must be remembered that the First and Fourteenth Amendments
safeguard the rights of free speech and assembly by limitations on
state action, not on action by the owner of private property used
nondiscriminatorily for private purposes only.
Whereas,
In the instant case, there is no comparable assumption or exercise of
municipal functions or power.
There is no sense in which social media, shopping malls, or news businesses constitute governments: the notion of "public space" is not what's relevant, rather, "government agency" is. "Unfettered access" requires the combination of "is or functions as a government", and "is public (not closed)".
As far as social media (a kind of private property) are concerned, there are recent cases involving what government officials may restrict w.r.t. social media, in particular involving accounts by government officials. Davison v. Loudon County, where a government official banned an individual from access to the official's Facebook page for 12 hours. One of the essential findings of that (lower) court is that the official operated "under color of state law" in maintaining the account, and as such, the banning was a violation of the First Amendment.
There is also a First Amendment case Knight First Amendment v. Trump in the works. The complain alleges that Trump's "account is a public forum under the First Amendment" – we'll see where that goes. However, these cases involve access-limitations by government officials regarding the official's primary means of disseminating policy.
If the First Amendment said (or was interpreted to mean) that anybody has unfettered access to a "public space" and social media constitutes a "public space", then Facebook et. al. could not have terms of service, and all accounts would have to be accessible to everyone – banning simply could not exist. The First Amendment right to freedom of speech is not content-limited to e.g. "matters of public concern" or "things about the government" – you are allowed to protest the actions of a private individual on public land.
