Are you playing by ear or from written music?

Start with simple tunes you are familiar with---ones that you can already sing or whistle or hum from memory.  Then make the violin imitate the sounds that you hear in your head.

I wonder whether this is what the OP is getting at:

The first piece I learned from the notes alone was The Ash Grove.  I started reading and playing it without first hearing it.  I'm not much of a reader, so I was plodding along, trying to play each note.  But at first I didn't get the tune. 

If you are reading a grammatically-complex English sentence, at first you see words and maybe phrases, but if you can't parse it, you can't make sense out of it.  You're trying to figure out how each part fits into one coherent structure.  Then after you match the words to your idea of English grammar, you get a good feeling, and you get the meaning.

Only after I was able to play the notes of The Ash Grove smoothly did something click, and I was able to see the overall structure in terms of what I might call musical grammar.  Then I got the "meaning" of the tune.  I think the best measure of reaching that point is that you are able to remember the tune in your head.

That's a pattern that basically repeats whenever I learn a melody for sheet alone.  After I master each note, and then master the phrases, the whole thing comes together, and I don't need the sheet anymore.

Yes, once you master the individual notes and appreciate the overall structure, you gain a lot by memorizing.  If you can translate the memory of the sound of a tune into fingerings, then you are an ear player.  And don't think that that is something magical or mystical.  It is a skill that can be gained by doing.

When you read, your eye sees the dot on the staff and your finger goes to the right place on the fingerboard.  Playing by ear, you hear the note in your head, and your finger goes to the right place.  So you see that the memory of the sound of the tune replaces the sheet.  Either way, your goal is not to memorize a sequence of finger positions.  Either way, you learn to translate to finger positions.  Either way you plod along at first.  But you keep doing it and it gets a lot easier.  And they are not mutually exclusive.

In playing a simple tune by ear, what you need to have memorized is the sound of the tune.  If you play a simple tune that you can already sing or whistle or hum, there is nothing more to memorize.  If you gain a lot of skill at the above-mentioned translation, you will be able to pull some old tune out of your memory, one that you've never played, and play it correctly the first time.  Does that sound unlikely?  I didn't expect to be able to do it, but I was able to, after about four months of playing, after learning maybe ten or twenty very simple tunes by ear.

It's better if you learn the simple tunes without using sheet, and it is best to stick to just one key, probably G, at first.  So it helps to be able to identify the scale of a piece and to be able to transpose.  If you don't know how to do that, just look for sheet music of very easy familiar tunes with one sharp in the key signature.  That ensures that you'll be playing in an easy familiar key.  Then use the sheet music only for the first line or so; sound out the rest of the tune without using the sheet, by trial and error.  Later you branch out to other keys.