Does Islam promote violence?

No, Allah says in the Holy Qur'an:

Say, "Come, I will recite what your Lord has prohibited to you. [He commands] that you not associate anything with Him, and to parents, good treatment, and do not kill your children out of poverty; We will provide for you and them. And do not approach immoralities - what is apparent of them and what is concealed. And do not kill the soul which Allah has forbidden [to be killed] except by [legal] right. This has He instructed you that you may use reason." [6:151]

At another place, He says:

And do not kill the soul which Allah has forbidden, except by right. And whoever is killed unjustly - We have given his heir authority, but let him not exceed limits in [the matter of] taking life. Indeed, he has been supported [by the law]. [17:33]

Allah also says:

Because of that, We decreed upon the Children of Israel that whoever kills a soul unless for a soul or for corruption [done] in the land - it is as if he had slain mankind entirely. And whoever saves one - it is as if he had saved mankind entirely. And our messengers had certainly come to them with clear proofs. Then indeed many of them, [even] after that, throughout the land, were transgressors. [5:32]