It all depends on your employer's policies. Typically, no employer is required to offer sick time. While we all need mental health days to recover, if the employer wants proof that you are sick, it is the employer's prerogative.
Here is another perspective: Unless there is a union, or a contract of employment, employees are all employed “at will”. This means that, without a union or a contract of employment, a worker can be terminated for any reason, or no reason, by an employer as long as they are not discriminating based on race, gender, national origin, religious beliefs or against another suspect classification, or unless the termination violates public policy. A public policy violation would include termination where the employee was voting or going for jury duty.
This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.