Can an employer fire me and/or press legal matters for calling someone a faggot on Facebook not a co-worker?

First of all, your employer cannot press charges regarding this posting. It has suffered no direct harm from due to it. However, that having been said, many companies have social media policies, which if violated, can result in an employee's termination. And even if no such policy existed, as an "at will" worker your compony could have fired you for any reason or no reason at all, with or without notice. That is unless doing so vilated the terms of an employment contract or union agreement. Bottm line, absent some form of legally actionable discrimination, a business may set the conditions of employment much as it sees fit.

No, there are no legal charges or lawsuits you could face if you did not in doing this identify yourself as an employee of this company, because without identifying yourself, any impact from your statement on the company's image is too speculative or causally remote as to sustain liabiltiy.
On the other hand, you can be terminated over this unless you have an unexpired written employment (or union) contract for a specified or definite term, which contract contains provisions preventing your termination for this reason. Without such a contract, you are an employee at will and may be termianted at any time, for any reason, including your employer not liking something you said or stated on your own time. Employees at will have no protections for their jobs.