Learned experts have given their valuable advice.
In your appointment letter your ex. employer must have mentioned period of notice for termination of employment from both sides and salary (basic/gross/CTC) to be paid in lieu of notice period. You may study your appointment letter yourself or show it to experienced/competent people in your family e.g your father, elders, or your acquaintances, or your lawyer/law firm.
Your company maintained silence on receiving your resignation letter. If at all they wanted to retain you they could have responded and requested you to stay for ....days and help them to complete.......assignment. They have maintained studied silence. While you followed up they sent you a bitter communication, threatening some legal action. Please note that the legal cell of any company is not any court of law or legal manager/HR manager of any company is not appointed governor by any state who shall enact any law and order any punishment to you. It is the nuisance of some jealous HR/legal employee of the company who has sent you a bitter email and is calling your current employer to disturb you.
You may handover the copy of your resignation by email to your current employer, and that should suffice.
If you fell it as appropriate you may send a communication to the concerned HR/legal cell employee of your ex employer and mention that his/her calls/communications to your current employer coercing them and you to give up your employment are unethical, unprofessional and illegal and you shall be constrained to advice your lawyer/law firm to initiate proper legal action in a lawful manner against him/her and their current and you shall also be constrained to inform all your known people to stay away from them and their present/future companies and in case they do not tender a written apology at once you shall be constrained to initiate a civil/criminal defamation against them at their cost and consequences. You may mention that you are contemplating to approach media, social and legal forums. Please use gentle language. If you feel you may take the help of your family or lawyer to draft a crisp and gentle letter. You shall see they shall stop.
The industry is full of such people. Be smart and learn to handle them.