Can an employee require me to finish college coursework at my own expense to get a raise for my current job?

I assume that was a typo and you meant "employer", not "employee."  Assuming there is no contract in place requiring your employer to give you a raise without finishing the coursework, and the employer is not discriminating against you based on a reason prohibited by statute (i.e. if you are female, he has given a raise to all males in your position without the need to do the coursework, but is applying that requirement only to you, or only to females) yes.  Your employer is under no obligation to give you a raise, and can make it conditional.