Employer refusing to accept resignation and refusing to give relieving letter/ experience letter

I work with a Non-Profit NGO. There was no contract signed, no offer letter or appointment letter. Due to some personal issues, I wrote a resignation letter with a three-week notice period, even though no such condition was mentioned while joining. My employer conducted meetings and tried to guilt me into staying for another two months. Meanwhile, after the resignation letter I applied and got an offer for another job. Since my reason for resigning was not the new job, I did not mention it to my current employers, fearing negative feedback to my prospective employers. Now my current employer is saying that they are not accepting my resignation, neither will they provide me with a relieving letter if I leave on the date I said I would leave, in my first mail. The new employers are asking for a relieving/ experience letter which the current employers are not ready to give, in spite of multiple requests. Can my new employers terminate me on the basis of not being able to furnish a relieving letter from the previous employers?