Does it count if you can shoot someone but only in one maybe 2[footnote]maybe it was 3, 4 if you count the bottles[/footnote] scenes and it's completely optional? Because I was thinking maybe Life is Strange counts.
Other than that, To the Moon and most of those games, actually lots of adventure games have no shooting, like I don't think you can shoot anyone in Broken Sword and I'm pretty sure you don't kill anyone, then there's arcady stuff like Marble Madness, lots of puzzle games, Sims, SimCity, actually a lot of simulation games, most sports games, too many racing games...
Yeah, there are actually a LOT of games that are not focused on violence, obviously though they are for the most part not action games but a lot of them are pretty mainstream.
Half-way around there I forgot I was supposed to mention games I liked so um... Well The Sims are fantastic games and I do love Marble Madness, I do recommend Broken Sword because it's fantastic, as for racing games, actually I like the most action packed ones but if you're willing to ignore tackling opponents until they die, F-Zero is perfect, that would be F-Zero x and GX, the Mode 7 ones are OK but I could never really get into them.
Besides that I really like Tennis games like Top Spin and Mario Tennis on the GBA, Poker Night at the inventory is a fantastic poker game too...
I feel like I'm forgetting something....
It was Harvest Moon, I absolutely loved the first one and Harvest Moon 64, those games are fantastic.
I also forgot Puzzle Agent, sure he has a gun but you don't shoot it, he shoots it but he doesn't kill anyone, that game is fantastic, telltale's magnum opus as far as I'm concerned.[footnote]Eat your heart out Walking Dead.[/footnote]