They would tear each other apart at this distance and become an orbiting debris cloud, eventually another ring of the giant.
I don’t think so. If both moons have similar masses (which is probably the case), they can be very close together without one destroying the other — although, as I said, the gravitational gradients would deform them. This is conceptually not different from very close binary stars like the Beta Lyrae primary pair.
(BTW, all Roche limit formulae I could find are derived with the assumption that
m≪
M and/or
r≪
R.)
How high is the gravity on these moons? Most moons I have visited so far have not enough power to tear another body apart.
That gas giant, though...
An icy moon 2000 km across (i.e. radius = 1000 km, quite typical for ED) surely has enough mass to disrupt a smaller, say Enceladus-sized (radius ≈ 250 km), icy moon if the latter came well within its Roche limit for a sufficiently long time. Their gravity may be weak, but the effects scale with the size and accumulate over time — that is what makes those moons’ shapes roughly spherical. (Incidentally, very small, irregularly-shaped moons tend to be more difficult to disrupt because they are held together by tensile strength more than by gravity; that’s how there can be a few of them inside Saturn’s rings.)
However, what we have in that picture is certainly two similarly-sized moons, so the actual Roche limit is likely to be less than the distance between them.
The gas giant, OTOH, appears to be at a safe distance.
