Yes it is possible to detect. Spam filters can classify spam material and it is used in data mining industries.
No, it is no way near ready for use for alerting when writing.
As Racheet said, this is an aspect of "Automated sentiment analysis" and "Natural Language Processing". It is possible to detect passive aggressive language in text. The problem is that such a system needs training. The system needs to know how you use language and needs to understand the context.
An example is the phrase "let it pour". A farmer may say "let it pour" in an email without referring to farming directly. This is a hopeful statement. A disgruntled worker may say "let it pour" in an email which could be taken as very aggressive. Additionally, This phrase may not have the same meanings in all countries.
If the emails from the farmer and disgruntled worker both use the phrase "let it pour" without stating they are a farmer or disgruntled worker, there is no context. There would need to be prior written text for the computer (or human for that matter) to work off.
This is not 100% accurate and the type of people using text editing programs expect the suggestions to be correct.
Additionally it would need to be trained for each new person. This training would involve the user stating if examples were passive aggressive or not. This would require 100+ examples of text. This is also hard for end users.