Is any kind of violence really required to have good action in a game?

cannot think bout it!
its very tough question... search google!😜

I think there's a problem, your OP has actual content.

Ambidot is pretty close to the problem with this sentence:

inevitably, there must be some kind of win or lose condition, or something to work toward, or it's no longer a game

Violence sometimes used as a motivational force, either to prevent it to happen to you, or to others - in many games you fight to protect others from the evil, from violence, or in almost any game the fail state is dying by the hands of the enemy. Dealing damage while avoiding it is the driving force behind Fighting games, beat them ups, shmups, mobas, fps ones,
If we consider the thrill of action it comes from the opportunity of failing and the promise of overcoming the enemies, either against environmental hazards or enemy forces. Enemy forces means violence, unless you play rock-paper-scissors with them :) and playing against nevironmental hazards defines one type of sport game to me, like Tony Hawk, or the games where you balance through a level with a bike, these games can lead to action-packed bits but afterall they are still "just" sports, in majority.

Why so Evil 1 & 2
I mean there was violence by me against the computer ... but nothing on screen.

bit.trip runner and many other rhythm games since the reason for action is moved away from violence.
maybe some clicker games

Thief series/Dishonored in "ghost" mode come to mind (no detection mode).

But in general no action game forces you to do anything, you can always opt not to "shoot" die honorably and quit. Or you can play the role you are given and play it by established rules, the same as any sports or board game.

depends on what is action for you.

In Soma you can't hurt anything but you can get killed. Same for Among the Sleep.
A Story About My Uncle is action without any fighting/enemies
Same for Event0 or The Turing Test but these two are more puzzlegames with a story.
Not sure if you hurt something in Papo & Yo and Contrast and if you would consider Brothers a Tale of two Brothers as action.

Life is Strange? Action or Not?

Depends of your personal definition of violence and the scientific definition of violence and agresion

It seems there are a handful of categories of action games (essentially, games where your skill/dexterity/reflexes are important) where you'll find non-violent games:

• Sports - already mentioned as being a special case/set aside. This may include driving and flying games, if they're based on motorsports or aerobatics (aerial version of motorsports) rather than being considered simulators.

• Abstract games - Tetris and Super Hexagon as mentioned previously, along with Snake, Breakout, etc. Thoth is a twin-stick shooter where you're controlling a geometric shape 'shooting' another and wearing it down. If the shapes represented ships or monsters, you could call the action violent, but they really are just shapes.

• Physics games - Somewhere between action and puzzle games.as they depend on judgement and precision control. Liquid physics (Puddle) and marble (Marble Mayhem) games might fall into this category, as well. You're exerting a kind of control that makes it an action game, but in place of a protagonist (person, creature, or a vehicle even), you're controlling a game piece, so even if it's destroyed, it's a thing rather than 'you.'

• Games where the 'violence' is therapeutic/redemptive/purifying - Super 3D Noah's Ark, is a re-skin of Wolfenstein 3D in which you try to use a slingshot to deliver tranquilizers to the animals on Noah's ark - the violence is bowdlerized (replaced with a safe/clean version). In Dustforce you clean your opponents, redeeming/purifying them (though I expect you can be killed by them). I'm sure there are many other games where you face characters that are antagonistic until you cure them or exorcise them, though I'm drawing a blank at the moment. Tron 2.0 featured curing infected programs, though I recall there being conventional violent action to it, too. In most of these, there's the risk of violence against you, which might rule out this whole category.

Most endless runners have some sort of violence close on the heels of their protagonist, but I think Race the Sun and Robot Unicorn Attack can qualify as non-violent, though as others have repeatedly said, it depends on your definition of violence - they ultimately end in a crash, but it's due to misadventure not conflict. First-person platformers like Refunct and Deadcore have that skill-based gameplay that I think qualifies them as a kind/subset of action games (though non-violent parkour games might overlap with sports games).

hard to say. most games that people mention here are either violent in the sense you described, or they are not categorized as action games (like Tetris, Portal...).

what about Tony Hawk's Pro Skater? it might be considered action, but completely non-violent. i think i solved your puzzle! yay! ^^ unless you count that as sports...

Little Big Planet series may be a good example - you can consider it action packed, especially timed levels, yet there's not any direct violence involved at any point (well, you can consider "dying" when you screw up "violent" but I wouldn't go so far ;p)

I don't really think of Overcooked as violent, but you might consider avoiding mortal accidents to be so. More to the point is whether or not it could be considered "Action." There is certainly a ton of frantic action going on, but the game is not tagged as Action.

If only violent games fall into the category of "Action," then violence is required in such games by definition.

When it comes to family friendly multiplayer shooters I've heard Splatoon is pretty good but sadly only for Wi U or maybe as an alternative for PC Plants versus Zombies Garden Warfare on Origin.

what about Tetris? ;P
except those blocks are sentient

Sims 3 or 4 \o/

again in regards to the main point I'm just going say...
100% a game doesn't need to be violent for it to be fun and there are countless options when you remove the requirement of action. For example its impossible to die in fez you just teleport back to wherever you fell off of.and a whole bunch of point and click games don't have any.
however as soon as you turn that sentence into "action games" then give no guidelines to what that covers I don't believe you can have a good action game without violence.

Pepsi Man.
https://www.youtube.com/watch?v=leGGsdfp4xo

I always wanted a sequel to the game that never came :(

Doom

Sure you can, a bit depending on your exact definition of action. You could make something like Super Monkey Ball, or you could make a game about some extreme sport. Heck, you could just turn the good old action mechanic of running up and hitting someone into running up and hugging someone, which would be mechanically pretty much identical, but not violent.

Viscera Cleanup! Totally not about violence, you are just, uh, doing chores, keeping stuff clean, usual stuff... :P