It's just a visual rendering artefact from having perfectly aligned and overlapping shapes.
You can see the same thing by perfectly overlaying a couple of rectangles:
You shouldn't see the same on a raster export. This is the same exported to a PNG:
The solution is simply not having perfectly aligned and overlapping shapes. So you build in to your shapes an offset one way or the other (either expand the top shape outwards or the bottom shape inwards). You can do this with a stroke, offsetting the path (Object → Path → Offset Path...), transforming the object... it doesn't matter how, as long as the shapes (or the visual appearance thereof) have some overlap and don't perfectly align.
In your example the white shape isn't needed anyway; it's just a side effect of how the artwork is constructed. So on your final artwork you can get rid of it... The easiest way is to expand everything (Object → Expand Appearance / Object → Expand) and use Pathfinder to merge everything. Leaving you with something this:
You say you already tried expanded everything and merging the paths, but there must still be paths where you see the lines; there's no way for them to be there without something being there to create them...
An easy way to check is to enter Outline View (cmd+Y).
You should see this:
If you still see paths where there shouldn't be then you need to go in and manually fix them. Manually cut the paths and rejoin them or nudge the points so both shapes overlap and use Pathfinder on the individual pieces. Again, it doesn't really matter how, just that you remove the unnecessary overlapping paths.
This is a common problem. It's the reason for the hidden notches in Hillary Clinton's logo and plenty of previous Q&As:
