To the extent government may impose negative consequences on speech, that speech is not free. This employer's opinion, which you quote, is not necessarily what a court would conclude, if this deputy were to challenge his suspension as unconstitutional. It is also simple-minded pap.
Of course our legal system not only tolerates but protects bias by private persons against other persons because of their race, religion, or sexual identity. Bigots do not enjoy the protections of the Bill of Rights any less because they are bigots. Any person is free to hate any other person or group of persons on those or other grounds, and, in his private relations, at least, to discriminate against them. He may call the members of this or that group all sorts of vile things, as long as what he says does not violate defamation laws or create a clear and present danger of violence, and he may refuse to associate with any of them. Despite the efforts of people who loathe the Constitution and want to enforce political correctness, this is still a free country.
What if a cop, on his own time, wrote a letter to a newspaper calling President Trump a "deplorable knuckle-dragger," a "fake-Christian stormtrooper," and "a Beverly Hillbilly--white trash with money"? Surely disparaging an individual in such an "offensive and repugnant manner" would betray "a deep and troubling bias." Considering that police officers are "charged with upholding the values of fairness and equality in the discharge of their official duties," and that at least some of the people in the community served by the officer must have supported Mr. Trump, wouldn't the officer's statements in the letter show that he clearly possessed "an inability to fairly interact with all citizens" in the area the police department served?
We have come to a pretty pass in this country, when harsh invective against a President by one of his detractors now seems to constitute lese-majeste.