Although there certainly would be differences of a sort between the way Paul spoke to Jew and Greek, no indication exists that would suggest that these differences were of a methodological sort.
As an example of the sort of differences that did obtain, consider that the apostle named names when he spoke to a Jewish audience about Jewish figures and events, but refrained from doing so when speaking to a Greek audience about those same things (e.g., the apostle does not explicitly mention by name Adam or even Jesus in Acts 17 even though he does refer to them). The manner or form of speaking was different, but the matter or content was not.
Similarly, and to the point, Paul did not formally or directly quote the Scriptures or cite chapter and verse when speaking to a Gentile audience, but he nevertheless did reason with them from God's revealed truth, whether that which is found in the Old Testament, and there are many allusions to key O.T. passages in Acts 17, or the deposit of truth that belonged to him as an apostle of Christ: "That which you worship in ignorance, I proclaim to you."