Classically, strong women were those who endured. City of Women, a text by a Cristine de Pizan was full of early "strong women," and then you could look at the more recent Jane Eyre. While some women in City of Women were fighters, it wasn't often. Jane Eyre wasn't violent, but sheesh did she endure some terrible luck!
Women who endured were oppressed. They mostly accepted their station and found their identity inside what was socially acceptable. These often were career roles limited to things associated with a maternal instinct: teaching, perhaps healing, etc.
Looking at the medieval construct of the interdependent woman, the woman had to be devoutly religious and sexually unavailable (nun, anchoress). Having any sexual identity went against female independence, because sex and marriage were associated. This association means that there was a power play in the sexual act, coming from the wife being in an objective role, a submissive role, to the husband. The wife was owned, so exploring her own sexuality was to put herself in a position off lesser power aqnd authority.
This power structure was shattered by the pill in the 60's. If pregnancy was dissociated from sex, then sex could be had outside of marriage and a woman need not place herself in a submissive role in order to explore her sexuality.
Clothing that allowed a woman to express her sexuality became affiliated with independence, from mini skirts to halter tops and daisy dukes.
Only, they of course backfired, and we associate these fashions now with the exploitation of women.
Women in SciFi Shows, in order to be depicted as "strong" were increasingly dressed in uniforms (Star Trek, Babylon 5, SG-1, Aaryn Sun in Farscape). Their outfits, while perhaps sometimes tight about the chest and hips, were honestly unisex. They were dressed the same as the men.
So what about strength?
If women were increasingly not forced to accept a submissive role, then to be strong they had to do more than merely endure.
What do we tell a little boy on the playground who's being picked on? Give the bully a punch, at least if the same media we are taking issue with in their depiction of women can be trusted to accurately depict the common American parent. We tell the kid to stand up for themselves.
Some parents tell their kids to use words. Thanks Mom, love you much, but that didn't work out to well for me in the 4th grade.
Just speaking isn't seen as strong in our culture. Sass isn't seen as strength, unless it can be followed by action. That goes for men as well as women.
So violence, being an action, is associated with strength --across gender. So if you want a woman to be seen as strong, she needs to be associated with action.
Action is not necessarily violence, it can be political maneuvering, social manipulation, or an ability to make hard calls in a career or leadership capacity.
Then, women, like men, can be considered strong for their ability to either maintain or discover a moral compass despite the trials they are forced top undertake thanks to our plot, story line, etc.
We are a competitive society, so in order to demonstrate that a woman is strong, she, like a man, needs to beat her foe. This can be literal or not.
But, as a competitive culture, we also use violence in a very allegorical way.
Our worst enemy is ourselves.
We fight our demons.
Keep your friends close, your enemies closer.
He beat out his best friend for the medal.
In a literal woman-warrior scenario, the fight is often allegorical. We use SciFi to explore ourselves, the same way any art is used. But thanks to the capacity to build full worlds and histories, the issues we take on can be much more complex, ongoing, and subtle, than is the majority of our literary counterparts.
In order to answer the question: Can a woman be strong without being violent?
I would say, yes, but she has to be able to obtain authority, power, and distinction among her community. Strength and some measure of cut-throat-ness is associated across gender designations. But we have examples of women who can do more than endure, we have examples of women who can stand up for themselves and rely on their own sense of morality to triumph as heroes (heroines) in movies, fiction and shows.