Making a Will

Since your father is stated to be critical and admitted in hospital, he is mentally unfit to execute any will. 

May God bless him with speedy recovery and long life.

Explore the possibility of video recording if he is able to speak and decide.  You can obtain his signatures on the will in presence of Hospital doctors.  Let the will be signed also by two different persons, preferably from Medical field and known to him.  He has a fundamental right to state about his last rites. You can also obtain a declaration from hospital authorities stating that he is mentally fit to take decisions.

You may contact a local reliable and competent Lawyer to help you. If your father is although not able to sign, is able to understand what he is doing, it's better the his  thumb impression taken on t Will in the presence of Registrar, seeking visit to your father's bedside and complete the Will Registration. One condition is that your father should be in a clear state of mind and understanding what he is doing. This fact the Registrar should accept and permit Registration of Will. 

He may be terminally be ill and not capable of signing properly. That does not necessarily mean he is mentally incapacitated. My father, who was a member of the Constituent Assembly of India, could not sign properly in his old age; the Bank Officer came and recorded his oral statement and I was authorised to sign in cheques on his behalf. He dictated a will and was written by my our neighbour, he affixed his thumb impression and it was witnessed by the neighbour who wrote the will, a Doctor who certified that he was mentally fit and only having a shaky hand and another neighbour. It was not even Registered. It was Registered post his passing away. It ws probated at the High Court of Madras. So it is possible

The testator is stated to be "unfit" to depose, hence attempt to execute a will would be waste 

The meaning of "unfit" must include mental incapacity