Oddly enough, the "cover it with rock" can be the problematic part, it creates traps for pockets of water and dirt. The fabric also helps trap washed dirt, sand, and bits of leaves. Next thing you know you still have weeds (or even trees and brush). And if you used coarse rip-rap, now you can't mow it either. I have seen a lot of trees and brush growing in rip-rap lined ditches here, though it does at least seem to help keeping the grass down.
I don't think your weeds will be coming from the bottom up. I think they will be coming top down, on the air or water, and get deposited on top of your fabric and grow just fine. Being in a drainage channel, they will be well watered.
I don't think even slabs or pavers would help you, other than limit growth to the seams. I suspect regardless of what you will use you will still end up either mowing or hand cutting brush. If you spray stumps afterwards, you could at least keep it from regenerating as fast...?