I have played just about every edition of D&D after the white box, and I play 4E. I'm currently running a 4E campaign that intentionally breaks the Gamist expectations of the rules. I occasionally play in a "Western Marches"-style Darksun 4E campaign that very much adheres to the Gamist expectations of the rules.
Gamism as I define it is a type of play in which the group of players are all interested in being tested by tough challenges and showing what they're made of (as players, not characters). It doesn't preclude creating story or developing a character (personality) or making tough moral decisions or immersing in a fantasy world, but these things take a back seat to the primary goal of play: winning challenges.
Can you play D&D 4E in a non-Gamist way? Yes, obviously. You have to break the reward cycle to do it, though, and the 4E reward cycle is very strong. Breaking the reward cycle means ignoring some of the rules, and if you ignore too many rules, are you still playing 4E?
The Reward Cycle
When I talk about the reward cycle of 4E, I mean the internal currency mechanism that rewards players for a certain kind of play. The basic cycle is this:
Players make some tough choices about their characters. The first thing they do is create a character, but this also includes choices made during play, and choices made between levels.
Players pit their characters against all manner of dangerous challenges. This rewards them (the players) with feedback about how good their choices are.
Players reward each other with feedback about how good they think everyone else's choices were. "Great job, Adam! Your fighter totally saved the lives of everyone in the party because you blocked that narrow door!"
Players earn experience points (XP) and treasure. These are types of game currency that make their characters stronger. Stronger characters can face tougher challenges! This changes play dramatically. This changes the types of decisions players will make during play.
The end of that cycle feeds back into the beginning again. Players take the feedback from the game play, feedback from the group, and the in-game currency, and apply that to new choices. The new choices include new character options when leveling up, and new tactics and strategies "in the dungeon" (or wherever).
Note that my examples are all combat and treasure, but this applies for other parts of the 4E game, like skill challenges.
Also note that I leave the roleplaying stuff out. While this is an important part of play for most people and the rules do talk about roleplay, the rules don't specifically include roleplay in the reward cycle. You earn XP by defeating monsters, by overcoming challenges, and by completing quests. Any house rule that awards XP for good roleplay must award enough XP to overshadow the normal XP awards to have any real impact, too.
Breaking the Reward Cycle
In my Saberpunk campaign, I basically just keep mental track of how many meaningful "scenes" the party finishes and award a new level to the group every ten scenes. Scenes include any kind of interesting conflict that the party faces. Combats are obviously part of that, as well as skill challenges, but some of these scenes are resolved with simple choices and roleplay. I give out treasure at a much slower rate than the book suggests.
So I cut the cord at the currency end of things. However, the leveling is still there. That means the power level still changes and the players are rewarded with shiny new powers and new challenges to face. Those things are a strong pull for players, who inevitably want to see their characters' new powers in action.
My Saberpunk campaign is, as I expected, drifting back to Gamist play. I'd intended a more Sim game built around a cyberpunk mood and setting. That stuff is still there, but it's getting lost. Our sessions are moving back to the "one or two big combats" that I was trying to avoid. The players have actively voiced wanting those combats and I won't deny them the fun they want.
At the same time, 4E doesn't entirely do what I need it to do. The Sim style of play I want begs for some additional character skills (in this case, Perform for the bard, and an Espionage skill for the wizard -- though Insight will do in a pinch). We just hack those in with a house rule. I'm using the Obsidian Skill Challenge system, too, with great success. I use a Blood Points rule to reducing whiffing and make combats less deadly.
This is just my latest attempt in a handful of failed attempts to drift D&D away from Gamist play. Even older versions have a reward cycle much the same as 4E's. My 3.5E campaigns trended the same way, even with significant XP awards for playing to specific, player-written goals/beliefs. While the XP awards were significant enough to make killing stuff less important, the reward of leveling and getting new character abilities was more powerful, drawing players into combats just to see their higher-level character do his or her thing.
The game gives you an asskicking character, and players want to kick some ass.
What is 4E anyway?
How far can you bend (or ignore) the rules of a game until it's no longer that game? There's no one answer to this question, for sure, but my personal feeling is that you should be able to bring in an average 4E player off the street and, without telling him what you're playing, he should recognize the game.
4E is, at its core, about 50 pages of actual "framework" rules. Most of the rest is rules-by-exception stuff: character race and class lists, magic item lists, equipment lists, skill lists, ritual lists, and monster lists. Some of the rest is play advice. The core of the 50 framework pages is the reward cycle. That includes: rules for creating a character (minus race and class definitions), fighting monsters, handling skill challenges, awarding XP, and leveling up.
When you start ignoring the core stuff, you start drifting away from 4E. I replaced the leveling up system with my own system (which happens to be very similar to a variant in the DMG2, though). I replaced the skill challenge system with Obsidian. I tweaked some rules to make combats more fun (like Blood Points, and like letting characters make a skill check as a Move Action instead of a Standard Action).
However, if Joe 4E Player came into my house on alternating Thursday nights, he'd recognize what we were doing as 4E. It might annoy the crap out of him though, since I totally de-emphasize the getting-into-combat stuff, which is the bread and butter of a lot of 4E games.
When I hear people say that they run games where there are never combats and no one rolls any dice, I wonder why they still insist on calling that D&D, let alone D&D 4E. What about it is 4E? I could bring a GURPS Fantasy character, or a D&D 1st Edition character, or a Rifts character into that game and play, right?
If "playing 4E" means using all the rules as written, I'd say that it's nearly impossible to stop playing it in a Gamist way and still have fun. You'd have to build characters, fight monsters, overcome skill challenges, earn XP, level up, and not care about that reward cycle. Let's say you're interested more in reinforcing the cyberpunk-infused-D&D tropes of the Saberpunk world than kicking ass. You're still fighting monsters. You're still earning XP. You level up a few times, and now your character definition includes a bunch of new powers. To use them, you need to level up more. You don't want to die, either, so you start applying the best tactics you have. Maybe you choose powers that make better sense as a build option than a character-development option. You're headed back to Gamism.
But very few people play using all the rules. Say you're like most games and you have a handful of house rules and you ignore some other rules, like I do with Saberpunk. It's still easily recognized by the average 4E player as "4E" and not some other D&D edition or some other RPG. The rules-as-written even tell DMs it's okay to tweak things. So is playing "4E" the same as playing 4E? I don't know. If you have to modify or ignore the rules to get some kind of non-Gamist play out of the system, is it really fair to say you're playing 4E? It's a philosophical point, and I grant that it doesn't have an easy answer.
Can you house-rule 4E so that it supports non-Gamist play and is still recognizable to 4E players? Sure. I recommend starting everyone at a higher level, tossing out XP and leveling altogether, deemphasizing the combats, and focusing on the kind of play you want. But now that you've thrown out the core of what makes 4E a D&D game, why didn't you just use a different ruleset to begin with?