This is not really a question one can know the answer to for certain without playing the game, but if I were to speculate, I don't expect dragon shouts to be necessary if you don't do the main quest, because they're primarily related to the plot about being the Dovahkiin and slaying dragons, so I wouldn't expect them to be necessary if you choose to ignore that aspect of the game. But whether one can complete the main quest without using shouts is another question, since they are a plot point in the main quest, it may be that some situations will require dragon shouts, I can't be sure, only time will tell.
Regardless, if I don't play the main quest, I might not use dragon shouts, but for characters who do, I think I'll use them even if I don't use any of the magic skills, because as I see it, dragon shouts aren't really the same thing as the magic schools, for one thing, dragon shouts are an important part of being Dragonborn, whereas magic skills are what makes a mage, so if I'm the Dragonborn, it's natural that I'd use dragon shouts, but if I'm not a spell casting character, I wouldn't cast spells.
Rumors: you absord the dragon souls, by doing that allows you to do dragonshouts.
I think it's already confirmed you absorb dragon souls, and this is connected to learning shouts, although it seems like the messages we get on how this ties into the matter and works with the thing about walls. Perhaps absorbing dragon souls enhances your ability to use shouts or unlocks the ability to learn new ones, but reading the walls is necessary to actually learn them, I'm not sure.