Can you get by in this game with just melee, no bows or spells?

Dragons would be a huge, huge pain 'cause you'd have to wait for them to happen to land before you could touch them, but you could do it. And are shouts spells or not?

Yes the character would use shouts but that would be the only magic used.

Guess it could, though I find it more fun with archery with you could at least try ^^

Playing without archery is certainly possible. An armored warrior would do well no problem.

I tried going all melee, but I got to the point of frustration with dragons that I started into both archery and conjuration to help out. It was worth it IMO.