Would it be okay to trash my employee's wife on my companies twitter page?
Why do you want to embarrass your employee's wife?
I'm irritated by her because of this stupid comment she made about my mom.
Speaking ill about your employee's family members is unprofessional. Doing this will lead to problems for the company and it will cause your employee to disrespect you. Instead, you should talk to your employee about the actions that you found unacceptable.
WTF I am the CEO and I can say whatever I want.
Even more so as the CEO, you as the leader set examples in every action you take. If you want the company to be seen as reputable and professional, then you should simply block her on twitter or tell your employee to not let his wife insult your mom.