For Nickelodeon, Spongebob Squarepants is the only modern cartoon that I think will be considered a classic. Avatar won't be, as it wasn't terribly popular, and not really any different form any other action series we've seen before. Spongebob though, it's a giant and loved by people of all ages. Maybe Fairly Odd Parents at second place behind Spongebob.
CN's side of things, I don't think any will be aside from the DCAU stuff.
Disney's, well, maybe Kim Possible, but that's about it and iffy.
WB/Fox... hardly any kids watches those channels anymore, so I don't think so, aside maybe Yu-Gi-Oh being seen as the "00's Pokemon" but nowhere near as mainstream or popular.
Keep in mind the term 'classic' is subjective, it's not about quality or anything, just what's popular/memorable. Out of everything today, Spongebob is the only cartoon with real noteworthy popularity and hype. Anything else can be up to debate, but at best only to the fans of the show. Spongebob is the only definate answer I can see.