What does Acts 18:15 mean?

In the Roman Empire, there was no such thing as separation between church and state. Emperor worship bound every resident to a common religion. All gods were to be pre-approved by the Roman government. It was not legal even to privately worship a foreign god, as noted in Cicero's De Legibus, ii.8. So, when Jews from the synagogue in Corinth accuse Paul of encouraging worship of a strange deity, they think they've found a way to convince the proconsul to throw him out of town.

Paul's heart and habits have protected him. He is the apostle to the Gentiles (Romans 11:13), but his affections will always first be for the Jews (Romans 9:1–5). To that end, whenever he enters a new town, he first looks for where the Jews meet to worship; in Corinth, as in most large cities, that is the synagogue. After teaching there for several weeks, trying to convince the Jews and Gentile God-fearers that Jesus is the Messiah, the unbelieving Jews are so disrespectful he leaves. But some believe, including Crispus, the ruler of the synagogue (Acts 18:4–8).

Paul, a Jew, teaches in the Jewish synagogue, then teaches next door at the home of Titius Justus with Crispus, the Jewish ruler of the Jewish synagogue. It's little wonder Gallio concludes the argument is over religious interpretations, not any direct matters of civil law. The tribune in Jerusalem and Festus will have the same reaction (Acts 23:29; 25:18–19). Other sources say Gallio had health problems during the end of his installment as proconsul. He eventually left thinking Corinth was making him sick. But it was God who orchestrated Paul's relatively peaceful, lengthy stay in Corinth (Acts 18:9–10).