It really depends on what we are defining as a "classic", and that's mostly a personal decision, but..
For most people "classic" immediately refers to the 50s-80s: classic rock, classic R&B, classic cars, etc.. so in that sense, its an easy no. No one will ever be writing songs in the era ever again.
Furthermore, its hard to forecast what songs from the the past 20 years will stand the test of time and continue to influence people 50-70 years from now. You have to expect that this same conversation took place during the 60s as a lot of what we call classic were coming out for the first time. No one had any idea that James Taylor would be performing 'You've got a Friend' at every concert he ever did for the rest of his life. The same could probably be said for many other artists and tunes, I'm sure.
Generationally, anything this encapsulates a time and/or place for the majority of listeners can easily become a classic. Nostalgia is power like that.. Boy Bands and punk rock come to mind as musical vehicles that perfectly represent the 90's, for me at least. Sure it may not be everyone's cup of tea, or even mine right now, as someone who grew up in the 90's, you bet your ass I still have some Green Day on my iPod... it takes me back
haha
The best I think we can do is compare accolades and who spear-headed stylistic shifts in popular music. On that note:
Katy Perry's Teenage Dream is the first album to top Thiller for most #1's on a single album...
Do I have to remind anyone how ubiquitous Britney Spears, nysc, and backstreet boys were...
post boy-bands: John Mayer, Jason Mraz, Jack Johnson, etc led the pack with acoustic-y pop/alternative stuff in the early 00s...
From what I can tell, Mumford & Sons brought that kind style to the pop scene....
Daft Punk won best album of the year with their electronic/pop fusion which is coming in in a big way....
Green Day is easily the most recognized punk rock band ever, Basket case anyone? Time of your Life?..
Alanis Morissette's hits are still getting a lot of air time everywhere (Jagged Little Pill has sold more copies than Sgt. Pepper!.. in less than half the time!!!)
There are lots of possibilities, only time will tell.