I think a dice-less game is absolutely possible, though I can't help but feel pulled back by the need for something to replace them. For example, cards from a shuffled deck are a great way to get more or less the same randomized experience as dice. That being said, if you're thinking of whether it's possible to have a game without randomness as an element, then I'd give a tentative "yes" given that either a series of other kinds of calculations regarding player stats and pre-determined event stat requirements or simply have the players respect the in-game scenario and react realistically like on these forums (maybe with a voting system for in-person games, i.e. "No way could you have dodged that arrow. Overruled.").

Either way, it puts a lot more responsibility on the GM's shoulder to determine the outcomes of each of those actions. For instance, in your Marvel example, it sounds like the action point system is pretty cool, but from your description, it seems like the results of every action the players take is completely up to the GM to determine. That's where the dice are helpful, because it helps the GM point to an external force when something bad happens, i.e. "It's not my fault you died, the roll was clear!"

In video games, you're actually moving an avatar around, so it's easy to see when something hits you why you react the way you do, but in an RP, it's usually all just talk. But video games still take a lot from role playing games, because they take a lot more setup. My first thought for a non-random role playing game would probably be your best bet. You can allocate and earn stats (through level ups, items/equipment, etc.), and the outcomes of certain events with required minimum stats would be determined by simple math or the presence of specific kinds of items. The drawback of that would be that it requires more preparation of the various maps you'll be playing on (which of course is rarely bad)... And unless you're particularly good at improvisation and can decide on numbers and calculate quickly in your head, there's less room for improvising as a GM.

Anyway, those are my thoughts on the subject. TL;DR, dice in games simply exist to make everyone's lives easier while still offering some enjoyable complexity.