When an employer terminated employee, does employee has to serve notice period.

Could any one please provide your inputs in below situation.

HI,

I joined in a company left after 20 days. I informed them i won't be able to continue due to personal issues. They asked me to serve 3 months notice period or pay 3 months salary. I told them i can't serve immediately but i can serve after 6 months. They sent email to me saying that "you have been terminated". You can serve notice period after a month long leave and providing proper justification and that accepts it by company. or legal actions will be taken accordingly.

In offer letter it was there like need to serve 3 months or pay 3 months.

I didn't signed on offer letter. I just sent email as accepting the offer. And there is official bond other than offer letter.

Does company can proceed legal action ?

And Do we need to serve notice period once they terminated me ?

Do we need to pay 3 months salary if they take legal action ?

And i don't want relieving letter or any other settlement from company ?

As a general practice, appointment letters in every company, carries a clause on "notice period or salary in lieu of NP" under Separation clause. But notice period during probation period can be nil also in some companies.
As you left the company after 20 days from the date of joining, presumably you are during your probation period. Please check what is the separation clause during probation period.
However, as per your appointment letter, if NP is 3 months, irrespective of whether you are a probationer or confirmed employee, then in the case of termination by company/employer, employer should have paid you 3 months salary.

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