Dear Randhir Singh,
In normal circumstances, the newly appointed employee should have been given the letter of appointment within one or two working days. It is unprofessional to delay the issuance of the letter. However, in your case, this very unprofessional behaviour has inadvertently worked to your advantage. According to the provisions of the Indian Contract Act, 1872, an appointment letter constitutes a contract between the employer and the employee. Since the contract has not been formally established, neither party is legally bound to continue the relationship and either party can choose to terminate it at will.
Let us think the opposite. If the appointment letter had not been issued and the employee had stopped reporting for duties after twenty-two days, you as an employer would not have had any grounds to question their abandonment of employment.
As far the employee's threat of the employee to take legal action against you, you may you may disregard it. Before doing so, however, you may clarify your position and enquire under which specific law or act the employee intends to file the suit. Since you have not violated any laws, the question of a lawsuit does not arise.
As mentioned by Mr VM Lakshminarayanan, the employee is entitled to receive their salary for the 22 days worked and It cannot be withheld on any grounds. However, he is not entitled to receive the payment in lieu of the notice period.
Thanks,
Dinesh Divekar