That does not resemble any religious doctrine I know of. The predominant one in the US, Christianity, views punishment on earth as corrective in nature, ie for rehabilitation, and also views prisons as serving the purpose of separating bad actors from society so they cannot damage others. There is no place for retribution, vengeance, etc. American culture seems to be fixated with vendetta, but it is not religious in nature at all, just as it has nothing to do with their love of sports.
You need to learn more. It is warmed-over Calvinism and this strain is widespread in the US thanks to the influence of theologians like the late RJ Rushdoony. Look into the work of Chip Berlet on Christian Dominionism and its influence on politics, as well as Colin Woodard's excellent history book American Nations.
At the heart of abrahamic religion is the idea that sinners will suffer God's wrath, in this life or the next. Which in turn gives righteous people the authority to punish sinners, because they are carrying out God's will.
