Eh the blood in games is pretty weak, but I felt the same way when I was 13. Now the grittier and bloodier the better

(Being a 19 year old male helps though, yay for tetosterone!).
I think the reason people look down on those avoiding violent gmes is because a lot of non-violent games are really bad or no where as good as some more violent games. Or at least they think of the bad ones first like Barbie's Horses or something. Course this isn't true, SimCity series for example.
Some non-violent games I am thinking of;
Sort of Risk-like time waster game that is quite fun.
http://www.smozzie.com/fracas.html
The Europa Universailels and Victoria series, which have warfare, it is somewhat abstracted and especially victoria can be played without a lot of war. It is probably less violent than Civ. Crusader Kings is more about building a dynasty than controlling a nation (which can be a mental hurdle to get over).
Both morrowind and Oblivion's violence is pretty weak imo (more so morrowind) and both likely have mods to remove blood and can be played quite non-violently.
Portal, okay it has turrets with guns that shoot at you but I don't remember there being blood and the turrets are easily disabled. But it's a first person puzzle game really.
Possibly Darwinia, as the characters are all electronic sprites living in a virtual world (I can't get it to work on my comp though >.>)
Oregon Trail off Home of the Underdogs. Although it does require hunting.
Wikipedia article, but I doubt it will help much.
http://en.wikipedia.org/wiki/Nonviolent_video_game