Fwiw, “civil justice” is in fact used to refer to (many?) parts of the (US?) justice system where people entities sue each other. It’s true the phrase is less common (in part because criminal justice reform is a hot button issue) and I’m not surprised you haven’t heard it.
Here’s an example of a reputable seeming organisation using it this way. https://instituteforlegalreform.com/blog/what-is-civil-justice-and-why-is-it-important/
I think depends on context and what you’re trying to use the term for. In the original sentence that was quoted, I think “the justice system” or “law enforcement” would have worked though colloquially, some people would have misunderstood both as just implying the criminal side.
I think “the legal system,” “legal conflict,” or just “the law” also would’ve basically worked though each might have a slightly different connotation.