It is not permissible to expiate for a broken oath by fasting unless you are unable to feed or clothe needy people or to free a slave
Question

My father has made an oath to a relative not to tell his secret, but two days later we were shocked to know that my father broke his oath and told the secret. So now he has to expiate for this broken oath.
My question is: Can I fast 3 days on behalf of my father, as he is diabetic and says he will not be able to fast due to the hot weather and his illness, although he fasted the entire previous month of Ramadan? .
Praise be to Allah.The expiation for breaking an oath (kafaarat yameen) is, as Allaah says, freeing a slave or feeding or clothing ten poor persons. If that is not possible then it is fasting for three days. 
Allaah says (interpretation of the meaning): 
“Allaah will not punish you for what is unintentional in your oaths, but He will punish you for your deliberate oaths; for its expiation (a deliberate oath) feed ten Masaakeen (poor persons), on a scale of the average of that with which you feed your own families, or clothe them or manumit a slave. But whosoever cannot afford (that), then he should fast for three days. That is the expiation for the oaths when you have sworn. And protect your oaths (i.e. do not swear much). Thus Allaah makes clear to you His Ayaat (proofs, evidences, verses, lessons, signs, revelations, etc.) that you may be grateful”
[al-Maa'idah 5:89]
Thus it is known that it is not valid to offer expiation by fasting, except for one who cannot afford to feed or clothe ten poor persons, or to free a slave. 
See the answer to question no. 45676 for details on expiation for breaking an oath. 
Secondly: 
If your father is not able to feed or clothe ten poor persons or to free a slave, but he is able to fast, then he must fast, and it is not valid for you to fast on his behalf. The one who has fasted all of Ramadan is not incapable of fasting for three days, whether consecutively or separately. 
If he is unable to fast, then the expiation is waived in his case. 
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: If a person has to offer expiation for breaking an oath and he cannot afford to feed (ten poor persons) and he is not able to fast, then it is waived in his case because Allaah says (interpretation of the meaning): 
“So keep your duty to Allaah and fear Him as much as you can”
[al-Taghaabun 64:16]
“Allaah burdens not a person beyond his scope”
[al-Baqarah 2:286]
and the Prophet (peace and blessings of Allaah be upon him) said: “If I order you to do a thing, do as much of it as you can.” And he does not have to do anything else, because one of the established principles of Islam is that duties are waived if a person is unable to do them and he may do the alternative if an alternative exists, or something else if there is no alternative. If he is unable to do the alternative then it is waived altogether. End quote from Fataawa Noor ‘ala al-Darb. 
Whatever the case, your fasting on his behalf is not valid, because he is the one who has to offer expiation for breaking his oath. 
And Allaah knows best.