The answer to that question is no, at least according
to the U.S. District Court for the Eastern District of Virgina,
which has ruled that Facebook “likes” do not count as
constitutionally protected forms of speech. From the opinion in

Bland v. Roberts:

[Previous First Amendment rulings] differ markedly from the case
at hand in one crucial way: Both [precedents] involved actual
statements. No such statements exist in this case. Simply liking a
Facebook page is insufficient. It is not the kind of substantive
statement that has previously warranted constitutional protection.
The Court will not attempt to infer the actual content of Carter’s
posts from one click of a button on Adams’ Facebook page. For the
Court to assume that the Plaintiffs made some specific statement
without evidence of such statements is improper. Facebook posts
can be considered matters of public concern; however, the
Court does not believe Plaintiffs Carter and McCoy have alleged
sufficient speech to garner First Amendment protection.

At the Volokh Conspiracy, UCLA law professor Eugence Volokh
argues that the decision is just flat out wrong:

A Facebook “like” is a means of conveying a message of support
for the thing you’re liking. That’s the whole point of the “like”
button; that’s what people intend by clicking “like,” and that’s
what viewers will perceive. Moreover, the allegation is that the
employees were fired precisely because the Sheriff disapproved of
the message the “like” conveyed. I would treat “liking” as verbal
expression — though it takes just one mouse-click, it publishes to
the world text that says that you like something. But even if it’s
just treated as symbolic expression, it is still constitutionally
protected, as cases such as Texas v. Johnson (1989) (the
flag-burning case) show.

Read Volokh’s full analysis of the decision
here.