most likely
The government can't compel people to some sort of speech under the 1st amendment. Forcing a company to host people is compelled speech by the company.
It is well established that the government can't compel a newspaper to host its messages as it wants. The key case might be Miami Herald Publishing Co. v. Tornillo 418 US 241 (1974). In this case, it was deemed unconstitutional that a newspaper would need to host speech of a political candidate the newspaper didn't like in the same amount it had used to disparage that candidate.
While the Miami Herald brought the newspaper into the line by the action of the newspaper, Wooley v. Maynard 430 U.S. 705 (1977) held that the state could not force any citizen to host its motto. Or for the matter, any message.
The State may not constitutionally require an individual to participate in the dissemination of an ideological message by displaying it on his private property in a manner and for the express purpose that it be observed and read by the public. Pp. 71717.
Forcing a public web page to host advertisement or speech from any government - or under the threat of the government action - is compelled speech and violates the rulings of Miami Herald, Wooley and other cases.
However, there is a little light for the government under PruneYard, Turner Broadcasting and Rumsfeld. However, all of them don't cut here: Turner Broadcasting was about a service provider for radio that did not host its own speech. PruneYard is a shopping center that doesn't host its own speech and is only useful in California as there is a California constitution issue. And Rumsfeld dealt with military recruitment, which always is special.
A similar Florida law was deemed to be very likely unconstitutional by the (federal) Northern District of Florida (Injunction Text)
Addendum
A joint lawsuit by NetChoice & CCIA was filed against Texas on 22nd September 2021 (Complaint), asking for a preliminary injunction. NetChoice puts its filings on their website.
Further reading:
01st December 2021 Update
Indeed, the relevant parts of HB20 were put out of enforcement via injunction on December 1st, 2021, reasoning that:
Social media platforms have a First Amendment right to moderate content disseminated on
their platforms. See Manhattan Cmty. Access Corp. v. Halleck, 139 S. Ct. 1921, 1932 (2019) (recognizing
that “certain private entities[] have rights to exercise editorial control over speech and speakers on
their properties or platforms”). Three Supreme Court cases provide guidance. First in Tornillo, the
Court struck down a Florida statute that required newspapers to print a candidate’s reply if a
13
newspaper assailed her character or official record, a “right of reply” statute. 418 U.S. at 243.
[...]
In Hurley v. Irish-Am. Gay, Lesbian & Bisexual Grp. of Bos., the Supreme Court held that a
private parade association had the right to exclude a gay rights group from having their own float in
their planned parade without being compelled by a state statute to do otherwise. 515 U.S. 557, 572–
73 (1995).
[...]
Finally, the Supreme Court ruled that California could not require a private utility company
to include a third party’s newsletters when it sent bills to customers in Pac. Gas & Elec. Co. v. Pub.
Utilities Comm’n of California, 475 U.S. 1, 20–21 (1986).
HB 20 compels social media platforms to
significantly alter and distort their products. Moreover, “the targets of the statutes at issue are the
editorial judgments themselves” and the “announced purpose of balancing the discussion—reining
in the ideology of the large social-media providers—is precisely the kind of state action held
unconstitutional in Tornillo, Hurley, and PG&E.” Id. HB 20 also impermissibly burdens social media
platforms’ own speech. Id. at *9 (“[T]he statutes compel the platforms to change their own speech in
other respects, including, for example, by dictating how the platforms may arrange speech on their
sites.”). For example, if a platform appends its own speech to label a post as misinformation, the
platform may be discriminating against that user’s viewpoint by adding its own disclaimer. HB 20
restricts social media platforms’ First Amendment right to engage in expression when they disagree
with or object to content.
For these reasons, IT IS ORDERED that the State’ s motion to dismiss, (Dkt. 23), is
DENIED.
IT IS FURTHER ORDERED that Plaintiffs’ motion for preliminary injunction, (Dkt.
12), is GRANTED. Until the Court enters judgment in this case, the Texas Attorney General is
ENJOINED from enforcing Section 2 and Section 7 of HB 20 against Plaintiffs and their
members. Pursuant to Federal Rule of Civil Procedure 65(c), Plaintiffs are required to post a
$1,000.00 bond.
IT IS FINALLY ORDERED that Plaintiffs’ motion to strike, (Dkt. 43), is DISMISSED
WITHOUT PREJUDICE AS MOOT.
