This^ was my first thought on the quote (in the op), that I’m not sure there’s a universal enough understanding of ‘meditate’ to give a definitive answer to whether it would put an end to violence. It’s kind of like saying lifting weights will make a body stronger and therefore prevent injury- but a layman picking up some weights and lifting them without a more thorough understanding of how muscles work together is likely to actually cause injury instead (by making select muscles stronger, thereby creating an imbalance and setting the weaker muscles up for a more severe injury). And while this is a wild oversimplification- this is how I perceive the kind of ‘meditation/prayer’ that leads to more violence…..that it creates stronger tunnel vision, rather than building mindfulness and making a person more aware of their conscious choices (and the effect those choices have on their external environment). Some religions actually require tunnel vision in some regards- a blind faith in something which actually imposes violence because it invalidates the experience of anyone outside that particular (and arbitrary) belief set, and having ‘faith’ requires they impose this violence without questioning whether or not it is ‘violence’ on another being; prayer, in this regard, is antithetical to the kind of ‘meditation’ which the Dalai Lama is referring.
And at the op: I’m not sure I believe it would eliminate violence, but in the specific context in which the Dalai Lama means ‘meditate’ I do believe that it would significantly reduce violence. As he intends the meaning, essentially it’s becoming aware of the violence in oneself- noticing that which sets off the impulse to hurt or invalidate the experience of an other, and remaining mindful of putting compassion above that impulse. I do believe in the Buddhist precept that violence begets violence, and the best way to get rid of it is for people to become mindful of coping mechanisms they use to deal with it so that they don’t impulsively contribute violence in return when it crosses their path.