Is this an "at will" employee? That is, there is no employment contact or CBA in place? If that is the case, you don't need a reason. You can simply say, "Thanks, but your services are no longer required."
The caveat is that you may not terminate someone for a specifically prohibited reason, such as due to the employee's age (if over 40), race, gender, national origin, handicap, color, religion, handicap, pregnancy, etc. Nor may you terminate someone for exercising a protected right, such as taking FMLA, reporting for jury duty, filing a WC claim, making a good faith complaint to a government agency and so forth.
You certainly may terminate an employee upon suspicion that he is not acting in his employer's best interests - and doing personal business that is in competition with his employer absolutely qualifies. That said, good employee relations practices dictate that you should be reasonably certain of your suspicions before taking action. The law does NOT require that you have proof of misconduct or even any evidence of it. Since you may terminate for no reason, suspicion alone is sufficient.
I am just suggesting that a responsible employer will investigate their suspicions to the best of their ability to make sure they're making the proper decision.