To be fair and balanced, two months later I must do the logical followup to If arenas gave no gear, would people do them?
As with the arena post, I'd say the answer is "Yes and no, somewhere in between known as 'a lot less.'"
Can we test this?
The upcoming badge changes will make heroics into the new old arenas: unrated places to gradually build up high-quality gear. Will people leave raids for heroics? People left arenas for Naxx when that became the easy source of gear. Or will people use heroics "as intended" and use them to more quickly get ready for raids?
Perhaps the the question can be better phrased as "If there was an alternate system of advancement, would people choose it over raids?" Obviously that depends on the system and the person, but heroics, arenas, grinds, and buying accounts have clearly shown that some people will choose small content, PvP, grinds, or RL money over raids as a way to move ahead.
I do have a small fear that people will gear up from badges, look at early Ulduar loot and choose to wait for later bosses with upgrades; but then the lack of early raiders will make it harder to get to the later, rewarding bosses. This would confirm that some people do raids purely for gear rather than the challenges or some amount of innate fun in the process of raiding.
From a design perspective this might be a pointless question. If a game is based on gear, it makes sense that the relevant challenge would give the gear for the next one. In other words, raids must give gear. If they did not and we used purely a badge/heroic system, then you'd have a system where people are forced to spend a great deal of time not raiding in order to prepare for raiding. Whether it is gear or a thousand gold of consumables for old Naxx, grinding to be able to raid is just not fun.
In the end I always come to the same conclusion: A lot of people would, a lot wouldn't, and alternative character advancement is part of what makes WoW popular.