As scary and tragic as it may be to hear one's dad is ill, your DH needs to realize this isn't about him. Your DH isn't the one who is sick. Your DH's role is one of supporting cast not as the star. He should follow his dad's lead and not try to push his own pace and preferences.
I'm not sure where you are getting the idea that DH was being selfish or a drama queen, and I'm honestly a little offended by the accusation. He was distraught that his father, who is only in his late forties, might potentially die. How is calling him to find out if he is okay trying to "railroad" him? If your parent announced on facebook they had a terminal illness, wouldn't you try to contact them? You are being way to harsh. He was in no way trying to make it about him. He was worried about his father!
If my dad tells me something via Facebook that concerned me, yes I'd try to contact him but I would not call. I certainly would not call repeatedly. I would send him a note via FB or an email. Because, as I already stated, I truly feel one should reply via the same means they were contacted. IMO your FIL made it abundantly clear he did not want to speak verbally about his illness. So yes I absolutely think your DH trying to push his own preference to speak on the phone over your FIL's preference to handle it all electronically was a way of railroading over your FIL's feelings.
I'm sorry you feel offended by this. Clearly you and your DH have a different opinion. But that doesn't make your opinion right and mine wrong (or vice versa) both opinions are valid. But since your FIL is the one who is sick and perhaps facing death, I think your FIL's preference - no matter how distasteful you or your DH feel about it - is the one that takes precedence; if you FIL didn't want to talk about it on the phone IMO your DH was wrong to call, especially more then once, trying to get him to do so.
Is there an etiquette rule that says you must respond in kind to the communication you receive? So, if I receive a letter via snail mail, I am not allowed to call a person up or email them, I have to write a handwritten letter back? I just don't think that is reasonable.
I understand your basic message, but I think the harshness of your post loses the message you are trying to convey. "I'm sorry you are offended, but..." reads as very condescending to the OP.
I don't think this is so black and white. I give Dad a pass too. But that doesn't mean that no one else in his life can't feel hurt or worried or upset, and they should just stand back in the corner, quietly, and say nothing. While I agree the Father's wishes take precedence, his loved ones are also going to be going through this with him. They may not actually have cancer, but from personal experience, the entire family lives with the illness.