There is nothing in the law that mandates when an employee must be termed; nothing in the law that says when an investigation must be completed, and, in 49 out of 50 states including CA, they don't even need a reason to fire you. They do, however, need a reason constituting misconduct if they want to contest benefits.
It should be noted, however, that the state, not the employer, determines what constitutes behavior sufficient to deny unemployment. I have had employees denied for unemployment when I didn't even contest, because the state determined that the employee's actions warranted denial of benefits whether I contested or not.
By all means appeal. I'm not saying not to. No one ever prevailed in an appeal without filing one. But you need to have a realistic understanding of what the employer is likely to respond.

