Is it legal for an employer to deny unpaid time off?

IF your company is large enough to be covered by the Family and Medical Leave Act (FMLA) and IF the leave is to provide medical care or help the recuperation of the employee himself or his immediate family, and also IF the employee is requesting leave for that purpose, you'd have to grant it. I believe the cut off for FMLA leave is at least 50 employees; this would be something to discuss with your employees general counsel, HR consultant, etc. to see if the law applies to you and how to comply with it.

Otherwise an employer is under no obligation to allow unpaid leave. Whomever gave you the advice about the Employee Handbook actually has it backwards: IF the employee handbook stated that there was a right to unpaid leave, then if the handbook lacked any of the protective or limiting statements usually put into it (e.g. "policies may be changed at will"; "nothing in this handbook creates a contract of employment"), then it's possible the employee handbook would be taken to create an implicit contract guaranteeing that leave, at least until the policy is changed. However, where there is no agreement as to leave, not even anything in the employee handbook that might constitute an agreement or contract, then the employer is free to deny leave and to fire an employee who tries to take it. An employee without a contract is an employee at will: the employer may discipline, demote, transfer, terminated, etc. such an employee at any time, for any reason, and also has essentially free reign (subject only to obey wage and hour laws, don't discriminate, obey workplace safety regulations, etc.) to set work conditions and terms.