Dear Nayan,
First of all, not providing the appointment letter has made the situation tricky and diplomatic. It can be used as a double-edged sword from both sides (employer as well as the employee). It will be treated as if he is not aware of the terms and conditions of your company and employment, then how can you take action against him? Also, if he didn't sign the letter, there is no employer-employee relationship with you. However, it is a very delicate situation that can also overlap onto you, the employer.
Secondly, an absconding letter can be issued only when there is no reporting to duty without any intimation. But in this case, as I believe, he is coming to the office or reporting to duty but not following the company processes and rules.
Thirdly, the employer has reserved rights to terminate the service during the probation period subject to following certain rules and having a valid reason. Probation is given for the purpose of understanding whether the selected person is suitable for your organization or can perform well. Based on this, his employment can be confirmed.
Even though the termination of an employee during the probation period or post-probation is proscribed in "The Bombay Shops and Establishment Act" (as I believe this act is applicable to you). Section 66 states: Notice of termination of service:
No employer shall dispense with the services of an employee who has been in continuous employment for not less than a year without giving such person at least thirty days' notice in writing or wages in lieu of such notice. For less than a year but more than three months, at least fourteen days' notice in writing is required, or wages in lieu of such notice. This notice shall not be necessary where services are dispensed with for misconduct. For the purposes of this section, "misconduct" includes absence from service without notice in writing or without sufficient reasons for seven days or more; going on or abetting a strike in contravention of any law in force; and causing damage to the employer's property.
Therefore, I would advise you to give him the appropriate notice as mentioned above for the termination of his services.
Best regards,
[Your Name]