I believe the melody is in the core of everything we call music. But has it to be a sequence of tones that can be sung by a voice?
Let's have a look: Everybody knows the famous Prelude in C by Bach.
Can it be sung?
Hardly, or let's say, this requires some really serious vocal skills:
And it's definitely not exactly the way we think a melody should look like, isn’t it? Or why Charles Gounod was adding his Ave Maria on top of it and treating Bach's notes rather as a accompaniment, a beautiful though?
However, I think the Prelude is still a melody in itself!
It has a line of wonderful chord connections and it has true talking points as every good melody. What could be the most striking talking point of this chord melody? Maybe you guessed it already: The long pedal point right before the end.
If you take this away – the whole line of this beautiful trip of chords will crash as with every good melody where you'd steel a part away or add unneccessary 'talking points' that don't really belong.
Line of chords, hm, if this can be enough, maybe a beautiful line of instrumental colors will do fine as well! Just listen to this example of a sound timbre melody (German: "Klangfarbenmelodie") by Arnold Schönberg called "Farben" (colors), written in 1909!
Or a line of nice ocean wave sounds, maybe? But wouldn't this be just an atmosphere? Well, that's exactly the point:
If you feel a line, probably someone took you purposely a certain way through the sound territory he has choosen for you.
If you don't, if you hear just ocean waves, no matter in which order they crush to your ears, you may just sit at the beach and enjoy a beautiful sunset.
Either way, you are set!
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