https://www.etymonline.com/word/diabetes
The root word is not "di" (which isn't "die" either way). It's "dia".
https://www.etymonline.com/word/di-
None of these are "die". What you're describing would likely make the root word of "diabetes" to be "die" but "die" is too recent of a word to have many words using it. Most languages don't draw from English words; most English words draw from other languages.