Is the concept of a verb-subject complete sentence a cultural/linguistic invariant?

Both the concept of 'sentence' and 'complete idea' are a result of the pedagogical grammar tradition.

The fact that simple Subject-Predicate clauses are given as examples in the European linguistic tradition. If they chose, imperative sentences like 'Hand me the keys!', the question would not arise.

The notion of 'complete thought' is really just heuristic for identifying the minimal semantic core of a sentence. E.g. "A man walks." and "It rains." vs. "A wise man walks very quickly when it rains." But it is clear that the constituent ideas of the above sentence are not complete with respect to the speaker's intention, nor are they complete when compared to the subjectless command "Sit!".

In fact, you could argue that "Sit!" expresses a more complete idea than something like "a man walks" which rarely occur on their own.

So, as you can see, the question does not even make sense in English, let alone in a comparative perspective.

However, setting the complete idea thing aside, it seems that the subject-predicate structure of a basic declarative clause is present in all languages although the grammar of this can be quite different from English (as in ergative languages). See Dixon's 'Basic Linguistic Theory' for more details. But it would be dangerous to try to make much more of it than the fact that by definition declarative clauses are there to say something (predicate) about something (subject). [Edited to add] This does not mean that the simple declarative clause is in some way basic to language or the only way to express ideas in language. But in many contexts (logic, philosophy of language, school grammar) people are used to talking about language as if expressing simple predicates was what the rest of the system was growing from.