As an employer you're able to make the decision to fire an employee at will
for just about anything beyond the realm of protected characteristics
(i.e., being over 40, belonging to a certain religion, race, etc.)
But it's a bad practice
Although there's no law against it, firing employees simply for asking for a raise isn't a good business practice. You want to keep employees who put their best efforts into their job, and are willing to go the extra mile. As an employer you want to incentivize high-quality work performance, and part of that may involve being willing to negotiate a raise for employees who are truly working above and beyond their current pay-grade.
Explain to your employee why you will or won't give them a raise. Is it that the bottom line simply won't allow it, or do you think that their work performance isn't going above and beyond their current wage? Whatever the case, your employee should appreciate your honesty, and having a better understanding of what it may take to earn a raise can act as a motivator.
The following responses to being denied a raise can be proper grounds
- The employee becomes angry or makes threats
- The employee's quality of work performance diminishes
- The employee becomes a counterproductive influence in the workplace.
Although possibly permissible, firing an employee for asking for a raise isn't a good idea unless their approach to asking or reaction to your response is unprofessional, or otherwise demonstrates poor employee conduct.