Praise be to Allah.
Smoking is something that is prohibited in Ramadan and at other times. This has been explained in fatwa no. 10922. It is also one of the things that break the fast, as has been explained in fatwa no. 37765.
If it is proven that your father has broken the fast during the day in Ramadan, by smoking or otherwise, then he has committed a grave sin, which is one of the major sins.
Ibn Khuzaymah (1986) and Ibn Hibbaan (7491) narrated that Abu Umaamah al-Baahili (may Allah be pleased with him) said: I heard the Messenger of Allah (blessings and peace of Allah be upon him) say: “Whilst I was sleeping, two men came to me and took hold of my upper arm, and brought me to a rugged mountain. They said: ‘Climb up.’
I said: ‘I cannot do it.’
They said: ‘We will make it easy for you.’
So I climbed up until I was at the top of the mountain. There I heard loud voices. I said: ‘What are these voices?’
They said: ‘This is the screaming of the people of Hell.’
Then I was taken and I saw people suspended by their ankles, with the corners of their mouths torn and pouring with blood. I said: ‘Who are these?’ They said: ‘These are the people who broke their fast before it was time to do so.’”
Saheeh Mawaarid ad-Dam’aan (1509)
Al-Albaani (may Allah have mercy on him) said, commenting on this report: I say: this is the punishment of those who fasted then broke the fast deliberately before the time for breaking the fast had come. So how about those who did not fast in the first place?! We ask Allah to keep us safe and sound and grant us well-being in this world and the hereafter.
Please see also fatwa no. 38747
What you must do is be wise in dealing with this difficult problem. If you think it most likely that your father will benefit from you speaking directly to him, then speak to him about that and explain to him that what he is doing is haraam.
Otherwise, try to convey that message to him indirectly, such as playing a clip that he can hear about the ruling on smoking during the day in Ramadan, and that it is one of the things that break the fast, or place somewhere that he can read it a written fatwa that explains that, and that speaks of venerating the month of Ramadan and the seriousness of violating its sanctity.
If you think it most likely that negative consequences will result from speaking to him, then do not hasten to open the subject with him, until there is a suitable opportunity to do so.
Such prohibited matters are well known among people, and no doubt your father is well aware of the prohibition on smoking whilst fasting, otherwise he would not have done it secretly. So the issue is not one of explaining to him something of which he is not aware, or establishing proof against him, because the matter is clear without any need for you to speak and advise him. So all that is left is for you to examine the situation from all angles, which you know better than anyone else, because you know your father and his character. So weigh up the interests that will be served by speaking to him, and what negative consequences for yourself or for the family may be expected as a result of that. If you fear some obvious harm or negative consequences that outweigh the benefits, then you do not have to tell him what to do or what not to do, or advise him or exhort him at all. Your only duty to him is to pray that he be guided and set straight, and to look for a suitable opportunity to advise or exhort him, even in general terms, and indirectly.
At all costs avoid watching him or spying on him, because that is not permissible, in addition to the fact that if he feels that you are spying on him, that will lead to him being put off and not accepting your advice. We have previously discussed ways of calling Muslims to adhere to the fast in the month of Ramadan; please see fatwa no. 50745.
And Allah knows best.