Raid content remains some of the most advanced in terms of its design - as the recent series of dev watercoolers points out, it's seen a great deal of iteration over the years of its existence. It sometimes seems to be the most time consuming in terms of development. Would it be feasible to simply not do it? And what would we replace that kind of content with?
One possibility would simply be to replace it with more five player dungeons. If several 5-mans could drop with each content patch, they could be deployed at certain difficulty levels to provide a form of progression for players who want that -- but it seems like you wouldn't be saving a lot of time that way. Boss encounters would still need to be designed and tested, as would trash -- and ultimately, the tuning to create a sense of progressing from easier bosses to hard would also be time consuming. Still, it's certainly possible.
More world events would also be possible. More world bosses, more questing (max level questing has already been mentioned as a goal for Warlords) and perhaps even events like the Battlefield Barrens we saw in Mists of Pandaria. With the coming of garrisons and content accessible through them, there's a means to keep max level players occupied that has nothing to do with raiding anything -- other kinds of content could be designed and implemented in much the same way. One could easily imagine an entire expansion with patches that opened up new world zones for level 90's (so, similar to patches 5.1, the Isle of Thunder in Patch 5.2, and the Timeless Isle in Pacth 5.4) and new five mans to replace the raids, as well as a few world bosses for larger groups. It's not impossible to imagine. The question becomes, then, would it be worth doing? How intrinsic is raiding to the World of Warcraft experience, given that a great many players never get to do it at all?
If you'd asked me at the end of Burning Crusade my answer would have been far less equivocal -- servers basically had something like 10% (at most) of their communities involved in raiding, the game certainly could have handled no new raids for a long while. But with the various ways raiding has been made accessible and even integrated into the design of the world as a whole, it's also become a part of WoW's identity. Raiding is, for some of us, what we do in WoW. Whether you're just hitting up LFR on all of your alts during a week, or you're in a small guild bringing as many friends as you can find and trying flex, or you are raiding hard modes (and can now bring players from other realms) raiding is something more people do now - it's gone from an activity only a few players could engage in to one that a great many more, in one form or another, can make a part of their gameplay.
Can World of Warcraft exist without raiding? Could an expansion launch without raids, perhaps go the entire length of its existence without them? I'd have to say yes. If I could come up with alternative content offerings for one, I'm sure the people at Blizzard could do a far superior job. The difficulty is, would a WoW expansion without raiding feel like WoW at all? Would people stay, and keep playing, if the expansion came and went and no raids came with it? That I can't answer. I know that for me, the game is about the raid game. I'd be much less likely to play it without them. But I suspect I may be a minority there.