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From what I remember reading, a sentence in Japanese must either end with an
adjective, a copula or a verb, if we exclude sentence ending particles and
inflections. This being the case, I'm under the belief that "元気" by itself
does not constitute a grammatically correct sentence in modern Japanese. Is
this correct?

Furthermore, I'd like to know how a Japanese sentence is rigorously defined
and what components are necessary.
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question
grammar
Is "元気" a grammatical sentence in modern Japanese?
Even from a grammatical point of view, 元気 is a perfectly fine Japanese
sentence.

> From what I remember reading, a sentence in Japanese must either end with an
> adjective, a copula or a verb, if we exclude sentence ending particles and
> inflections.

This is more or less accurate. Here, though, we need to see that 元気 falls into
the category of "ending with a copula", but with the copula itself omitted.
"元気。" or "元気？" in its full version can be

>   * 元気だ。
>   * 元気です。
>   * 元気ですか？
>   * 元気です？ (Yes people generally add か but grammatically it's not
> **NECESSARY** )
>

It just happens that だ can be omitted, and です, although not omitted because
it's respectful language (omitting is considered disrespectful or casual),
there is nothing grammatical that stops these copulas from being omitted.

Therefore, "元気", whether a statement or a question, is a completely
grammatically correct Japanese sentence. If you have time, I suggest reading
[this chapter of Tae Kim's Guide to Japanese
Grammar](https://guidetojapanese.org/learn/grammar/stateofbeing) which does
explain on this topic.
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answer
Yes, I believe it's a Japanese sentence (sometimes called a minor sentence or
a [sentence word](https://en.wikipedia.org/wiki/Sentence_word)).

Looks like many English speakers were [taught at
school](https://japanese.stackexchange.com/a/48864/5010) that a sentence
_must_ contain a predicate (a verb or a copula). However, as a native Japanese
speaker, I have never heard such a rule from Japanese teachers in schools in
Japan. Things like `元気。`, `はい。`, `おはよう。` or `わ！` are always perfectly
grammatical Japanese sentences to me. Simply, any valid expressions delimited
by periods (or question/exclamation marks) are grammatically sentences. Some
consider these "incomplete" sentences, but they are not even incomplete to me;
there is no "full version" of こんにちは, for example.

If you look at the [Wikipedia article about
sentence](https://en.wikipedia.org/wiki/Sentence_\(linguistics\)), you can see
many linguists think a sentence does not have to contain a predicate.
Something like "Mary!" or "Wow!" are sentences to them.
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answer