By definition this is not an intentional crime or tort (i.e. civil wrong for which one can sue).
There are several standards of intent (also called mens rea) other than knowledge that one is committing a crime or intent to commit a crime, that are commonly applied to criminal offenses and torts:
- Strict liability
- Gross negligence
- Willful and wanton conduct
- Extreme indifference
Strict liability would be highly unlikely to apply to unintentional encouragement of violence. Usually, in the criminal context, it applies to traffic offenses, like speeding or drunk driving defined by blood alcohol content.
Negligence is often a basis for liability in a lawsuit or for other civil remedies (e.g. cause to fire someone from their employment), but is usually only a basis for criminal liability when a death or very severe injury results (e.g. vehicular homicide) or when the circumstances are such that there is a heightened risk involved in an activity (e.g. discharging a firearm, or treatment of a small child in one's custody). Even then, for criminal law purposes, liability is usually only imposed in cases of truly "gross negligence." One could imagine highly stylized fact patterns where gross negligence encouraging violence could give rise to criminal liability (e.g. gross negligence by the commander of a military unit under military justice) but this would be a rare and exceptional situation.
Willful and wanton conduct, and recklessness, don't require actual knowledge or specific intent, but do presume disregard for objectively obvious risks. Under the Model Penal Code, this is the default standard of intent that applies when no specific standard is articulated in a statute. Unintentional but reckless conduct encouraging people to be violent might very well give rise to criminal liability.
Extreme indifference (also sometimes called deliberate indifference) is an extreme form of recklessness in which one acts with total disregard for the consequences when one clearly knows or should know that serious consequences are almost certain, even if the precise consequences to whom or what are not known. This level of intent is often treated as equivalent to intentional conduct and would often give rise to criminal liability.
There are many potentially relevant federal and state statutes that could apply, but this is the general lay of the land regarding unintentional conduct.