Yes, the employer can file a case in civil court, not in criminal court, as it is not a crime. However, if you want to file a case against an employee for not serving the notice period, then I think you should first check the terms and conditions of the appointment letter given to your employees. Most companies include a clause regarding a shortfall in the notice period, which stipulates that any failure to serve the specified notice period or any shortfall period shall be deducted from the employee's salary.
In such a scenario, if the employee still owes dues to the company, the company can recover them, and the employer cannot file a civil case against the employee. If there are no pending dues or if they cannot be recovered from the employee's salary, you may send a formal notice for recovery. If the employee still does not pay, then you may proceed to file a civil case against them.
Regards,
Tushar Swar