Albert Barnes on Acts 18:15

Of words. A dispute about words; for such he would regard all their controversies about religion to be.

And names. Probably he had heard something of the nature of the controversy, and understood it to be a dispute about names; i.e., whether Jesus was to be called the Messiah or not. To him this would appear as a matter pertaining to the Jews alone, and to be ranked with their other disputes arising from the difference of sect and name.

Of your law. A question respecting the proper interpretation of the law, or the rites and ceremonies which it commanded. The Jews had many such disputes, and Gallio did not regard them as coming under his cognizance as a magistrate.

Look ye to it. Judge this among yourselves; settle the difficulty as you can. Comp. Jn 18:31.

For I will be no judge, etc. I do not regard such questions as pertaining to my office, or deem myself called on to settle them.

(++) "words and names" "Doctrines" (c) "look ye to it" Jn 18:31, Acts 23:29, 22:11,19