I hear it all the time from my colleagues: But that's correct, or? However, my colleagues are all German, and in German, one can turn a statement into a question by adding oder (which means or). Does this sound natural to a native speakers ears?
Or can be used to extend questions where there are other options not being mentioned that the respondent can fill in. Generally speaking, it cannot turn a declarative statement into a question. Examples of where it would be properly used are:
Do you want to go to the movies, or ...? [the mall]
Would tuna for lunch be ok, or ...? [no, I'll have salmon instead]
It is important to note that typically, or would only be used followed by a trailing ellipses.
More common are yes and no, which can be used to turn declarative statements into questions, such as:
She found $20 at the beach, no? ["didn't she" is another acceptable alternative to "no" here]
He had lunch yesterday at McDonald's, yes?
Either way, whether you use yes or or or no, such usage is generally accepted in speech and in informal writing, but not in formal writing.
waiwai's answer is very good and detailed, but I have something else I'd like to add. Judging from the question and some other comments, I'm thinking that the closest English construction to the original German is "or not".
So she found the shell on the beach, or not?
As you said, the "or" in german is use to change a declarative into a question. "So she found the shell on the beach" is a perfectly acceptable declarative sentence; in English the "or not" is appended to the end to change it into a question.
To use your original example, "But that's correct, or not?" would be a perfectly acceptable English construction. It seems to me that the only difference is that in German the "not" is implied, whereas in English it has to be stated outright.
So that answers your question, or not? ;)
As others have suggested, "right" can also be used in these situations, as can "yes" or "no" (depending on whether the statement was positive or negative). But this seems to be the closest construction to what you mentioned in the question.