It's possible, but you have to avoid the audience expecting action, which is kinda hard. It rules out characters like Superman, where people want to see Superman punch something, and a more introspective movie will be maligned for not fulfilling the promise of the character.
You essentially will need to focus the movie on an action-light character, and even then, even movies like Titanic, or Gone Girl have action, it's just not superheroic beat-em-up action. There are a few ways you can go with this that seem immediately apparent to me:
A) You can do a character who is not a classical superhero, that is a Lois Lane movie will be immediately understood to not be about beating up Brainiac.
B) You can do a character who is so god-like that it's impossible to do an action movie with him. A Dr. Manhattan movie would be inherently introspective, and people would understand that going in. I think Superman Returns was going for this, but for many who see Superman as an action hero, it was insufficient.
C) And then, in a similar vein you can do a character whose powers are not offensive, that is a Jean Grey movie, for instance, is understood that there isn't a lot of punching. There may be 'action' but the action is more dreamscape stuff, and may involve little to no 'combat' though then you're blurring the line between drama and combat, which, I'd hope, is the point.
Another important aspect is how a movie is released and promoted. A Hawkeye family film that is promoted years ahead of time with a budget of 120M could be seen as a disappointment. But a cheap and dirty style film, not unlike Deadpool. Also, the drama will have to be good because the action is considered good, otherwise it'll be a 'should've done what they're good at' thing.
For me personally, I'd love a Green Lantern movie in which punches are shown to be unhelpful and creative solutions are how things are resolved, but that's not too likely. I think Dr. Strange, in being focused on it's, imho, limited physical combat and foot chases instead of the much cooler spatial manipulation gamesmanship that it could have been was a lost opportunity. I think the resolution was much more in keeping with an action-light magic user like Strange, what with out thinking one's opponent's reality. I think there are other heroes whose conflicts could be restrained to more social and cerebral conflicts that are enabled or empowered by their powers, but none come to mind so quickly at the moment. I can say a Fantastic Four movie that ended in the same kind of beat-em-up at the last three would be disappointing, but I don't think there should be a beat-em-up scene in there, either. I think there are a number or properties, X-Men included, that would work, perhaps even better, as drama-centric with only one or two real action scenes in there. Certainly the Phoenix/Dark Phoenix Saga doesn't work as a beat-em-up, for instance.
But for those who really just have NO action, something like a King's Speech or a Wolf of Wall Street, where superheroics are analagized to a mundane job that the hero is simply trying to deal with purely from a dramatic standpoint, there's not many heroes that could get away with not showing off the action their powers promise, and indeed, that superhero comics are built on. Perhaps original characters could get away with this better, but the most important part is making sure the audience doesn't feel promised any action, and that, clearly, is no mean feat.
It would pwn though, and if done right could make some tremendous Oscar bait.
t:
