It sounds like the issue here is control. You are calling your father and family in order to be there as much as you can for your father. And you also want to know how he is doing and show you care. But for some reason, this has become about controlling how you communicate rather than just the fact that you call because you care.
The other issue is Michelle. The problem there seems to be her commenting on your conversations with your family. What you want to find out is why she feels the need to comment on your conversations. Is it because she feels left out? Or hurt by the treatment from your family? Or is she also seeking some sort of control over you? Once you know why she is commenting, it is easier to address.
One of the most important things being left out here is what you want. You are being put between Michelle, your family and their issues. Each side wants you to do something, neither of which makes you happy or addresses what you want or need. Yet both your family and Michelle care for you and love you. So why are your needs not being addressed here? It is important to focus on what you need to make this work because that is going to address the issues you are dealing with.
Start by talking to both sides. With Michelle, once you find out why she is commenting on your conversations, try to address it. For example, if it is because she feels left out, tell her that you will either include her in the conversations or let her know once you are done talking with your family what was said. That way, she does not feel the need to stay around and comment.
With your family, stand your ground. Decide what kind of communication you are able to do at the time, and stick to it. So if you decide to call one day and your family protests because it is not face time, you can say that it is all you can do now and if they are that upset about it, you will call back later. Or ask why it's so important that you called instead of face time, since your point in calling is to check on your father, not the method you used to contact them.
Also, if you feel you are talking to them too much, cut back. It does not mean you care any less if you don't call several times a day or however many times you call. You care just as much no matter what. And someone's medical condition does not change that much that it would require you calling each day. If something changes, your family will let you know.
The main point here is putting your needs first since no one on either side seems to be doing that. They are all pushing their own agendas. So you will have to push your own. You are not doing anything wrong by saying what works for you. And if they protest, make that their issue, not yours. Their protesting is more about not getting their way and losing control over you than anything else. But if it continues as it is, the only person who is going to be unhappy is you because your needs are being ignored.