I'm in agreement with the other responders that email is not the proper way to communicate but if this is how you do it best now, please do not stop. Though work hard at soon having a better verbal way that puts you two face to face. You could progress from email, to phone, to in person.
(Sorry the rest is a long read!)
As for your hubby, he is angry. Based on what little info you could give as obviously this is about a situation and you two have your lives together, I'd presume he is angry about his father's cancer diagnosis. His behavior while not appropriate could be expected and your response while also not appropriate needs to be more forgiving if only for a little while.
I'm certainly not shaming you as you reacted how most people would. You did what you needed to do to protect yourself and only what you knew. Again, my advice is limited to what is included in your question and so I do not know all of your hubby's personality.
Simply put you said your F-I-L was diagnosed with cancer and since about then, your hubby has been moody and stressed. The problem started with him being upset after a short wait time in the car in which you did nothing wrong. Only problem is that his current situation and having 5 minutes alone in the car to be silent and think... well this brought up alot of stuff for his already stressed self to process. You responded well by letting things ride and not jumping back at him when you heard his negative tone but when things dived further down because he is angry. A terrible feeling to have is anger as it is just a protective cover for a more insecure feeling that we are desperately trying to avoid. I'm guessing your hubby has fear behind his anger and when he jumped a second time is SAFE place with the one he loves, you walked away. This may be best for your relationship as again, I do not know your specifics. I'm thinking a hug and a comforting tone of "I love and I'm here for you" may have been needed.
So since his two attempts to get it out, even though not proper ways to communicate but natural ways to try to feel better by spatting with someone else, well you handled yourself well and now he still hadn't worked through anything. His approach was unproductive and he needs to find better ways to initiate processing but then he could use some help processing even if he starts wrong.
So with all that said, he started again what hadn't been finished in front of your family. As for your reason for not wanting others to know your business... semi understandable. You should not divulge your issues to your family as that makes them prejudiced against your spouse as few can separate their feelings at a later date but also, your family can't be ignorant to think you never ever spat. You don't need to announce it upon arrival but tension can always be felt and if addressed or something comes up then you simply acknowledge your a married couple working through a bump. It's no one's business the specifics and if pressed you tell them it's a little bump that occurs in all relationships as every relationship is working partnership of compromise.
Now the real problem started with his inability to communicate healthy with you and I send HUGE kudos for handling yourself so well all along, but the point the things went south was when you refused to go visit your F-I-L.
I always tell people seeking marital advice or even advice with disciplining kids, lets go back over the situation and see what steps happened that led to the problem and then go back through the steps and see what we could have done different at point A, B, C, D, and so on.
So going over your steps... You hubby could have not had a hostile tone upon your getting in the car, he could have not spatted off that rude comment in the car, he could have not had hostility at your parents home and he could have shut the door calmly.... but you too could have not been spiteful in refusing to visit his father. Now I am sure it was not your intent to be spiteful as you stated that you didn't want your fam or F-I-L to know you two were disagreeing but in his eyes you were paying him back and being spiteful. From what I can see here is that he made many mistakes but your response was far more severe and a slap in the face to him.
I'm not saying your hubby was right in his actions but I think he deserves a little sliding room considering how grave his situation became with his dad's diagnosis. I think he needs to have a better communication level with you and might find some counseling helpful. What is going on in his mind is obviously worse than what you are seeing. I hope you find a way to let go of his actions and see that in another time they may have been rude or disrespectful. but in this case they were simply frustration and a poor way to handle himself. He certainly didn't put forth the effort to disrespect you and he is probably hurting at how he was hurting inside but went to dinner with your family and yet you spited him and his ill father by not visiting.
His email is proof he felt you were there before but that this time he felt slighted. Again, this situation with his dad is severe and nothing like what you two have encountered in your past.
Since you two have somewhat communicated, you know that what you have been doing didn't meet his needs this time. Not your fault but since he has eluded to what he needs, you should provide that. This is what relationships are all about. You are not bad, you are learning! Learning is something that never ends as well we all change a little and different situations call for different measures.
So tonight, just apologize for the whole other night. Tell him you didn't know what to do and just didn't want things to escalate. Let him know you really wanted to see his dad but didn't think going over there both tense would help dad out. Let him know in hindsight that you wished you;d have gone as his going alone was probably worse for dad. Let him know that you love him and while you are new to such a grim issue with a loved one, you are willing to learn. That you are not a mind reader and will try to respond appropriately but that if you are not meeting his needs, he will need to ask. That in the future if his anger arises that you will call him on it and when you do say something like, "your angry and it's not really with me, what is it I can do to help because you can't treat me like this and I really want to work through this."
Please whatever you two do, keep communicating via email if that is the only way, but please, please for your sake and role modeling to your children.... learn to talk face to face.
Then lastly, don't beat yourself up over the past, but learn from it. I hope this provides some insight so you can both have a better communication style and that your hubby receives all the support he needs in such a trying time.
God Bless & Many prayers for your F-I-L and family,
-mb