Can an employer fire me for something I wrote on Facebook even though they shouldn't have been able to view it because my profile it set to private?

As a matter of fact your employer can fire you for on-line postings. In fact, if you are an "at will" employee (and most employees are) your workplace can fire you for this or any reason, even no reason at all. Employers have a great deal of discretion when it comes to setting the terms and conditions of employment. Absent a company policy to the contrary, or a union agreement, or employment contract prohibiting this, your employer's action was lawful. In other words no laws have been violated. 

Note: Another exception to the above is if there was some form of discrimination at play (although you did not indicate that to be the case). That would be illegal.