What can I do if my former employer withholds hours from my final paycheck?

1) If you had earned and properly used the personal day, the employer may not dock a day from your final paycheck because you used the personal day. Personal days are part of your compensation, if the company has chosen to provide them; therefore, it may not after the fact take those days away--which is what it is effectively doing, if it reduces your pay for having used one--any more than the employer could retroactively reduce your wage or salary.

2) If you had and correctly used the day, then to fire you for that may be wrongful termination, since by doing so, the employer is firing you doing something which you had the right to do and thus effectively denying you part of your compensation.

3) The law does not require personal days to be paid on termination--it is up to companies to set their own policies on this. The majority of employers do not pay personal days (even if they pay vacation days) on termination of employment. However, if it had been the policy of your employer to pay unused time  like this, the employer must  continue to follow that policy--at least until it announces a change in policy--and therefore pay you.

From what you write, it may be worthwhile for you to consult with employment law counsel to evaluate whether you would have a wrongful termination claim. Even if you don't, you could likely sue (e.g. in small claims court) to recover any money the employer owes you, such as for wrongfully docking your pay.