Justices Roberts, Scalia, Thomas, Alito, and Kennedy say yes.
Justices Stephens, Souter, Breyer, and Ginsburg say no.
Note that the case involved a prosecutor who made a fuss about what he thought was a defective search warrant. But it applies to any internal whistleblower. The Court has in effect instructed public employees that they must remain silent at work about whatever misbehavior their colleagues and superiors choose to engage in, or risk retaliation.
And it was precisely because the prosecutor kept his complaint within channels, rather than going to the press, that the Court found that he was speaking as an employee rather than as a citizen, and therefore was not protected by the Free Speech Clause. Because the plaintiff had only been doing his job his actions didn’t constitute Constitutionally protected citizen speech. Justice Kennedy helpfully points out that employers who want to encourage employees to speak up through channels can create workplace policies that do so. (Sometimes I think the Court ought to replace the inscription “Equal Justice Under Law” over its portico with “Just Visiting This Planet.”)
Note also that, allowing for substitutions on the Red team, this is precisely the lineup in Bush v. Gore. (Though Marty Lederman at SCOTUSblog, who’s as appalled by the decision as I am, thinks that Justice O’Connor might have come down on the side of the angels.)