I want to say that I may die because I am angry.
Can I say "I am angry to die" or "I am angry to death" to express the above?
Your first choice of "I am angry to die" does not convey what you want to mean. It means that you are angry that you are dying.
One idiomatic way is to use the construction
X is an adjective (or adjective phrase) and Y is a verb phrase.
Where X is hungry and Y is eat a horse.
So for your example that would be
Your second choice, "I am angry to death", is OK, but I don't think most people would use it.
Americans like my mother say things like I'm sick to death of your moaning! I've never heard I'm angry to death. It's understandable but not idiomatic. People don't die of anger unless they have a stroke or a heart attack because of it (Type A personality people sometimes do). So I'd suggest you not use it.
I'm angry to die means that you're unhappy about the fact that you're about to die, so unhappy, in fact, that you're angry about it and want to curse God and whoever or whatever else might have caused your probably premature death. I don't think it's the right expression in this context. That should probably be I'm angry about dying (too soon / so soon / so young / for nothing / because I caught a cold / etc.).
"Angry to death" exists in literature, though not much used today. To that extent, "I am angry to death" is grammatically and semantically acceptable to mean "I am so angry, I could die."
(Dr. Wes Bredenhof God is Right to Care)
No. Matt Ellen has given the usual idiom, but has not addressed your samples.
can only mean I am angry (about the fact) that I will die/am dying.
could mean what you say, but it does not exist as an idiom. There is an idiom sick to death, but note that it does not mean sick in its main sense, but only in the sense of fed up, i.e. suffering because something has happened too many times. Note also that there is no literal contemplation of death in this idiom.
Since your example "I am angry to die" has caused so much dissension amongst the users of this site, it would probably also confuse your readers. Why not use another expression that expresses the same degree of anger but is more readily understood such as:
I am furious!
I am ready to blow my top!
I am mad as hell!
I am so mad I could kill!