That doesn't sound too well in English - Some verbs are followed by the direct object and the verb in the infinitive (to stop), and want is one of them: https://learnenglish.britishcouncil.org/en/english-grammar/verbs/infinitive
That yields "We want that process to stop" (want + something + stop); that's different than "We want to stop that process" (In Portuguese, "nós queremos parar esse processo"). In the first sentence you want it to stop, but you're not doing anything about it; in the second you're implying that the group of people you are a part of wants to stop it directly.
I noticed this doesn't sound nice, but this being a Portuguese lesson, I just would like to understand if my translation is sufficiently grammatically correct.
Duolingo proposed this two sentences as correct: "We want that this process stops" "We want that process to stop"
Knowing this, "We want that that process stops" would be incorrect? Doesn't really make sense to me...
I see from one of your later comments that you thought "We want that that process stops" should be accepted based on the fact that "We want that this process stops" is already accepted. That seems logical but there is a problem.
The problem is that "want" isn't one of the verbs that can be followed by a "that clause" in standard English. See: https://learnenglish.britishcouncil.org/en/english-grammar/verbs/verbs-followed-that-clause
As confirmation see the warning in this grammar note: http://dictionary.cambridge.org/grammar/british-grammar/want
We don’t use want with a that-clause:
"I want you to tidy your room before the visitors come."
Not: "I want that you tidy your room …"
Both references are talking about British English, but the same seems to be true for American English. See this Google Books preview of "The Columbia Guide to Standard American English": http://tinyurl.com/ohtbfl9
want that followed by a clause, as in She wants that you should come upstairs, is Nonstandard; to plus an infinitive is the Standard construction: She wants you to come upstairs.
I know this is quite a technical objection, but hopefully you'll be able to remember the standard answer next time you meet this sentence, and it won't hold back your learning of Portuguese too much.
Thank you Davu for this clear answer.
But then maybe this sentence is not the best choice for a Portuguese exercise, given that the English sentence is more complex than the Portuguese one.
Now, maybe that a native English speaker would not fall into the trap but Duolingo is not only used by English speakers. ;)
Maybe change this error to a warning would be sufficient?
Many people have suggested that "close-enough" English grammar should be tolerated. I'm not part of the course team, although I know the course contributors already have a difficult job assigning yes/no decisions without the additional burden of determining which cases should be accepted with a grammar warning.
Maybe one day Duolingo will introduce a Portuguese course in your native language. You can already learn it via Spanish and there are reverse courses in Spanish, French and (soon) German. Good luck.