Possibly doable, but probably not worth it.
Assuming we don't count outright cheating through debug/value editors etc, it still would require probably a lot of save-scumming and quite some knowledge with will to exploit every opportunity and mechanics of the game available. And time, a lot of time. Some missions are tiresome and long with proper guns and whatnot - using just fists with no other items in any way at any point would make it a masochistic nightmare, and that assuming you have aforementioned decent power suits that would let you punch later enemies and deal any damage at all which you may be unable to get to with punching alone.
