The problem here isn't really about whether or not a moon could "skim" a ring, but whether or not rings could form around Earth.
There are three ways planetary rings can form. All three involve material coming inside the Roche limit of the planet. The rings of Saturn have a mass of about $3 \times 10^{19} \text{ kilograms}$, one one-thousandth the mass of Earth's Moon. The formula for the Roche limit shows that the Moon is 41 times as far from the Earth as its Roche limit is, and a less-massive body of the same density would have a smaller Roche limit, so if this object was the same density as the Moon, it would have to have gone really close to Earth. At any rate, the Moon would be too far away from the ring system.
Anyway, here's how rings can form, and why they might not around Earth:
- Accretion from the protoplanetary disk. This is simple. Matter from the solar system's protoplanetary disk ventures inside the Earth's Roche limit, gets torn apart, and becomes a ring. I don't think this would happen on Earth because Earth is not massive and never has been, and so would not pull a lot of material towards it. Sure, it could get the required one one-thousandth Moon mass, but would it come close enough to form a ring?
- A moon gets hit hard. Collisions happen, and they happened frequently in the early solar system. After all, that's how Earth got its Moon. If a moon got hit hard enough that debris was flung into space and somehow got inside Earth's Roche limit, it could turn into a ring. However, this would necessitate a second moon-like object (or protoplanet) being near Earth at some point in time. It's likely that it would still be here today - after all, only a tiny bit of mass would have been lost. True, an asteroid could have been smashed, but what are the odds Earth would have gotten the debris?
- A moon gets ripped apart by tidal forces. This is the most commonly-cited explanation for ring formation. However, it requires a moon to go pretty close to Earth, as we can see from the Earth's Roche lobe with respect to the Moon. Likely? Perhaps not, considering that Earth's gravity was relatively weak in the early solar system, compared to the gas giants.
I doubt rings could form around a planet like Earth.
Let's say rings do form. Could our Moon skim them? Well, remember that they would be near Earth's Roche limit - 1/41 the distance to our Moon! True, the Moon is receding, but it probably never was even close to being that close to Earth. If it was close enough, it would be torn apart by tidal forces. Shepherd moons (as Twelfth helpfully brought up) are exceptions, but they are small, and could be fragments of a larger moon that was torn apart, or broke apart in a collision.
Okay, let's go to the Moon as a shepherd moon. The reason that I assumed that the Moon could not be a shepherd moon is because of the mass discrepancy. Here's what I'm talking about:
- Mass of the Moon: $7.3477×10^{22}~\rm kg$
Some shepherd moons:
There are almost certainly many more.
The point is, these moons are tiny. They have no chance of drastically perturbing the rings because their absence from a portion of the ring has no negative effect. The Moon, though, is much larger, and will greatly perturb all objects in its vicinity.
So what if (and this is a $\Huge BIG$ if, and I see I've found an interesting feature in $\LaTeX$) the Moon is close to the rings? Then the outcome tends to be that the ringed Earth gets another ring.
The constituents of planetary rings (no, they aren't continuous, although that would be awesome) can greatly vary in size. From Wikipedia:
The composition of ring particles varies; they may be silicate or icy dust. Larger rocks and boulders may also be present, and in 2007 tidal effects from eight 'moonlets' only a few hundred meters across were detected within Saturn's rings.
So we're looking at objects ranging from a few microns in diameter to a few football fields in diameter (and it doesn't matter which side of the Atlantic you're on) - although most of these objects probably aren't big enough to pull themselves into spheres. The Moon, by contrast, is about 3500 kilometers in diameter. It will clearly be the most dominant object in the area.
Look at this unfortunately compressed image of Saturn's rings:
Image in the public domain.
The zoom on the link is really bad, but this is a good compromise. Long story short, the Moon is big enough to fit through some of Saturn's rings and nearly all of the gaps.