Dear Harikumar,
Technically, if the appointment letter is not issued, then the employer-employee relationship is not established. Since there is no contract between the employer and employee, a question of terms and conditions of separation from the company does not arise. In view of this, should you abandon employment at your will? The theoretical answer could be 'yes,' but there are practical things that one needs to look at.
If you abandon employment, then the employer is not liable to give you the "Service-cum-experience Certificate." If you do not acquire this certificate, then how will you show your tenure at this company?
Whether the appointment letter is issued or not, your current employment was a source of livelihood for you. Therefore, it is appropriate for you to leave the company on a good note. Our future is unpredictable. Our life is nonlinear. Who knows the situation may force you to come back to the current company. Therefore, prudence demands to keep the door ajar. In case you abandon the employment, then the door of that company will close for you permanently.
Thanks,
Dinesh Divekar