Praise be to Allah.
We put the following question to Shaykh Muhammad ibn Saalih al-Uthaymeen:
One of the Sunnahs with regard to marriage is that it should be announced publicly, with beating on the daff or hand-drum (i.e., celebrations), so as to distinguish it from fornication, which usually happens in secret. If the marriage contract has been drawn up in accordance with all the requisite conditions, then it is valid even if the family does not accept it. The issue of kafaaah or compatibility has to do with religious commitment, as Allaah has explained in His Book. A Muslim man can marry a Muslim woman, or a woman of the People of the Book (Jews and Christians) on the condition that she is chaste, but he is not permitted to marry a mushrikah (polytheist woman) or a zaaniyah (woman who commits adultery or fornication). He should, however, try to find a woman who is strongly committed to Islam. A Muslim woman is not permitted to marry anyone except a Muslim man. A man of the People of the Book does not qualify in terms of compatibility with her, and she should look for someone who is known among the Muslims to be committed to Islam and of good character.
With regard to the situation described in the question, we may note the following points:
The husband does not have to divorce his wife if his father asks him to do so.
The fathers rights are great indeed, and good treatment of one's family is a duty. If your father is suffering from heart disease, it is better if he does not know about this marriage. It is unlikely that his attitude has changed because it is based on the class/caste system and it is not easy for seniors to change their beliefs.
You have to make sure you understand your familys current attitude and to what extent they are prepared to accept your marriage to this woman. Perhaps they have heard that she has got married, and they think that she has married someone else, so they want to make you feel better (by saying they would accept her because she now seems to be safely out of the picture). Or perhaps they have heard that she has got married to you and they want to hear about it from you. Once you are sure that they have changed their attitude for real, there is nothing to stop you asking for permission from them and from your father to get married. If they give you permission, then this is what you want; if they do not, then just stay as you are now, so that no bad consequences will come to them as a result of them knowing about your marriage.
As far as repeating your marriage contract is concerned, we put this question to the Mufti Shaykh Abd al-Azeez ibn Abd-Allaah ibn Baaz, who replied with the conclusion that if the first contract met all the requisite conditions, and there are no impediments, then it is a valid contract. It should not be repeated lest that open the way to playing about with it. You should strive to please your family in every way you can and tell them that the matter has been concluded in an appropriate fashion. If there is genuine cause to fear for your fathers life, then it could be said that repeating the contract could be done on the grounds of necessity. And Allaah knows best.