Appointment letter was issued to me, I denied the joining date in it. Can they legally bind me to serve a notice period?

I joined an organization and called my manager after 2 days that I cannot continue because I got my dream company. I did not sign any document and just attended the induction. While an appointment letter was issued to me, I denied the joining date in it. Can they legally bind me to serve a notice period of 2 months or pay my salary.

You had accepted the appointment letter and joined the organisation, was there two days too. Though you have not signed any other document you are bound by the appointment letter which you have acknowledged. The terms of appointment letter only can clarify your liability. The very fact that you were there for only two days does not bail you out of this liability.
If the organisation is ready to leave you without paying you any sum for your two days attendance, that is the best alternative for you. You may try to persuade them to this proposal. As per the provisions of Contract Act 1872, the organisation can only charge for the breach of employment terms, i.e., the direct expenses incurred for you. So even if they resort to legal remedy, there is not much they can get from you.

Yes, it has got legal consequences, since you have accepted the offer and completed your induction. The word 'induction' means you have been inducted as an employee in the company, therefore all the terms and conditions of appointment is applicable to you.
The employer may ask you to discharge the notice period or pay the consequential amount in breach of contract. Your version of denial stands invalid immediately after your joining.
Now, you persuade your head and HR to relive in a graceful manner.