The cosmology issues, to me, are among the most interesting aspects of setting conversion. And with 4e, we have a doozy!
To me, the answer seems a clear no, the core cosmology cannot be used with Dark Sun intact. But I wonder if there are others out there who think it can be.
Secondly, almost as a thought experiment, I wonder what the most minimalist of changes to the 4e cosmology could be accomplished to get the core to work and not contradict anything. For example, while minimal in effort, I think merging the Black and the Gray together to become the Shadowfell would horribly contradict pre-existing work.
My brain's not up for the effort at the moment . . . too tired . . . 
The problem is The Gray and The Black fit the Shadowfell.
But How i do it: Shadowfell = The Black
The Gray is more a mist between Athas & the Planes that planewalker must travel to reach any planes. Between the Shadowfell & Elemental Chaos it is slim (like a -2 to ritual), it grows ''thicker'' between Athas & THe Astral, the Feywild, like a -10.
Note that planewalking ritual are extremely rare and a character can be lost in the Gray if he miss the roll by the penalty.
Loss in the Black cost 1 Healing surge for each PC who travel and the caster must redo the ritual (I rule that cost for Rituals in Athas are double, Enchant Magic Item is x10) with the same penalty.
For the parallel planes at least, there's one possibility that could add the adventure richness of multiple planes, while keeping Athas the cosmically isolated, perilous husk we know and love:
Others have suggested the basic idea of parallel planes as being different points in history; not so much parallel dimensions. My own spin on the principle is as follows:
In place of the Shadowfell, there are portals that take people of great power and knowledge 10,000 years into a bleak far-future - sort of a Black Age. The Sun is even darker, the surface is dead rock, haunted by the undead remnants of the last civilizations - survival here is a lost cause. But underground, life might still exist. The Rebirth Races have been replaced by cruel Shadar-Kai and squalid hordes of savage Grimlocks that see all flesh as meat to be torn asunder raw. They fight a bloody war against the descendants of the toughest mutated monsters Athas has to offer. Adventurers crossing over into this Black Age must bring with them everything they need to survive, but may succeed in wresting fabulous treasures of the future from dark ruins haunted by the shades of the once-living. Maybe even power and knowledge to avert this Black future.
In place of the Feywild, there could be portals and rituals that time-travel you back to the Cleansing Wars. Many people on Athas would risk anything for greener pastures to settle in, but perhaps this magic only returns you to the height of the Wars, when the lands where embroiled in unprecedented violence against desperate demi-human and fey races amidst the most horrific defiler magic yet seen. Heroes would have to dodge bloodthirsty armies and magical cataclysms to glean secrets and treasures that might save the Present. It's simply not possible to go back any further, to peaceful times. Perhaps once you've jumped back in time, you can't do it again unless you return to your own time first - and you can't go back any further than the most brutal years of the Cleansing Wars.
This would provide different venues for adventuring, keep Athas isolated, and give players a more profound appreciation of the World of Dark Sun.
I think you could have all this and still have the Gray as a barrier between Athas and the Cosmos.
Another alternative would be to allow magical travel to one or both of the Moons of Athas, as a replacement for Astral planar travel.