So I just said “that can't be right English” (meaning of course I didn't think it was correct English), and they told me saying “right English” (instead of “correct English”) was improper and I would get points off a test. So... Is it really “improper” or not?
You might find the arguments on this page to be interesting: What is the difference between "correct" and "right"?
The consensus there seems to be that "correct" refers to things that are provable, or factual, while "right" refers to opinions, or preferences. In other words, "correct" is objective (meaning there is external validation) while "right" is subjective (meaning that it varies between individuals).
In most cases "correct" and "right" are interchangeable, since there is often no difference between objective and subjective. For example:
She gave the right answer, that Nairobi is the capital of Kenya
This is both objectively and subjectively correct, since it can be both proven by external sources, and also is generally considered factual by most people.
The phrase "right English" is subtly different from "correct English". It implies that the person strongly that there is a right way to say something in English, which therefore means that anything else is the wrong way.
In some cases this may be true, but in many others it's purely subjective opinion. The perception is that, while there are certain general rules, there is enough subjective variation in English language use that "right" and "wrong" are not as important as whether a particular instance of English is "correct" or "incorrect" -- that it follows certain established rules.
To get to the point: it is possible to say, "That is right English", but it sounds both awkward and pretentious. The more common expression would be, "That is correct English".