I agree. It should be "you didn't write TO me". Duo's suggested translation "you didn't write me" isn't properly constructed english". It's a colloquial american usage.
I think that possibly every BrE speaker has weighed in with a comment that it is abominable English to omit the "to" in "write to me". So if that is your issue, your message is received. That does not disqualify it as a correct and acceptable variation. BrE is not the gold standard for Duolingo's translations and there are just as many BrE translations which sound simply awful to American speakers. Let it be, please.
Duo is very US-centric, true, but "you didn't write me" is not "bad" English, as was asserted. If anything both "you didn't write me" and "you didn't write to me" should be accepted