Am I actually contracting the phrase by omitting "good" from it, hence the need for the use of an apostrophe?
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You don't need an apostrophe if you just say "Night." If you clip part of the "good" and say "g'night" then you need an apostrophe. This is actually how the phrase is heard most often.
I sense your discomfort at just using the single word as a sentence.
That's a perfectly understandable and even grammatical expression. The "good" is understood, because it is such a common expression.