"That which" is grammatically correct in English, but very stilted and awkward.
It can be a statement: "you do that which is right" it can also be an order, in that case I would accept "just do that which is right" (without "you")
"fai" is not the imperative form of fare so it cannot be used as a command. http://italian.about.com/library/fare/blfare160a.htm
Because (tu) "fai" is correct alternative form for imperative of the verb fare.
This sentence has two possible nuances from my experience as a native, a statement or an exhortation/a comand:
(Tu) fai quello che è giusto ogni volta che c'è un problema.
Fai/Fa' quello che è giusto! (the elision of 'fai' is not mandatory).
In the written language it is very frequent to add " ! " to avoid misinterpretations in these ambiguous cases.
This isn't necessarily an imperative sentence. It's ambiguous.
"What do I do, Dad?" "You do that which is right, honey."
vs. "I don't know what to do." "I'm sure whatever you choose will work out, because you do that which is right."
No be careful guys, Craig Pickering is right:
what/that which = formal ciò che / informal quello che
So 'what' is correct here. It translates a double pronoun: neutral demonstrative pronoun + relative pronoun. The neutral pronoun ciò/quello is used speaking about undefined things and it's always invariable (no inflections for gender/number).
Speaking of definite things instead it's possible to use quello che only and not ciò che and you need to translate it with the one e.g.
Quello che ho comprato è il libro di Jim. = The one I bought is the book of Jim..
Also, in these cases quello che has the inflected forms quella che (the one), quelli che, quelle che (the ones), because we're not speaking about general things.
As for the rest, what also means che cosa/cosa/che, an interrogative pronoun/adjective.
fa' with an accent or fai are both used for imperative. You do is present tense. And quello or cio' che are both that which.
Do the right thing was accepted. Good to see Duolingo translating sympathetically rather than literally.
Can you say "Do whatever is right"? I think that's less stilted and awkward??
No, whatever is qualunque. But "what" alone is both less stilted and shorter. I can't remember the last time I heard anyone say "that which", but I sometimes read it. (Perhaps the writer's editor is a pedant).
I don't understand why "you are doing that which is right" was not accepted, and the correct answer indicated as "you do that which is right". It seems to me to be the same meaning of a present-tense verb.
In everyday English they'd be interchangeable, but the important thing is to nail down the tense used because it matters more when we're learning the grammar. Your first response is present continuous tense whereas the answer is just present simple (or imperative/command form if need be).
Nobody would say 'Do that which is right'. Everybody would say 'Do what is right' (or 'Do the right thing', which I gather is acceptable here and which I will now use, as it sticks in my craw to translate good Italian into horrible English).