I am a manager for senior living and I do not like the new supervisor, can I fire him solely because we do not get along?

Assuming that you have the authority to terminate employment, I don't see why couldn't fire this employee.  Most employment relationships are what is known as "at will".  This means that an employee can work for an employer, or not, their choice.  In turn, the employer can hire or fire someone for any reason or no reason whatsoever.

The only exceptions to this are: 1) If there is an employment agreement or contract (which governs termination); 2) If there is a union or collective bargaining agreement (which governs termination); 3) If there is an employee handbook or other official policy statement (which governs termination); 4) If discrimination is a factor (there can be no termination for reasons of race, religion, age, disability, sex, national origin).