These were Gentile converts to Judaism that Paul was preaching to in the synagogues. There were no Gentile Christians attending the synagogue of the Jews. Jews and Gentiles did not associate with one another. Gentiles did not ordinarily mix with Jews and so they did not attend synagogue with the Jews unless they were already devout converts to Judaism or they were in the process of converting to Judaism.
Paul did not preach to Gentile/Greek Christians in the synagogues. Paul preached to Gentile/Greek converts to Judaism in the synagogues.
Acts 13:43
And
when the meeting of the synagogue broke up, many Jews and
devout converts to Judaism (Greeks)
followed Paul and Barnabas, who spoke to them and urged them to continue in the grace of God. rsv
The Gentile converts to Judaism who were attending synagogue on the Sabbath began to believe in what Paul preached to them about Jesus Christ and so they followed Paul and Barnabas out of the synagogue after the synagogue meeting was finished. After the meeting they asked Paul questions about his preaching and soon they accepted Paul's preaching about Jesus Christ. As a result of this outpouring of God's grace, these Jews and Gentile converts to Judaism believed what Paul was preaching to them.
As soon as they came to believe that Jesus Christ was the way, the truth, and the life, they started to attend the Christian gathering on the first day of the week and they became catechumens/learners of the Christian faith.
Shortly after God knocked Paul down and blinded him and then restored his sight, Paul began keeping the 1st day of the week holy and he did this because he became a Christian! Acts 26:13-15, Acts 20:7
People did not change their nationalities when they converted to Judaism. A Jew/Hebrew is always a Jew/Hebrew by nationality and a Greek convert to Judaism is always a Greek by nationality. There were no Greek Christians who attended the synagogues of the Jews. Only Greek converts to Judaism and Greeks in the process of converting to Judaism attended the synagogues of the Jews.
Acts 17:1-12
Now when they had passed through Amphip′olis and Apollo′nia, they came to Thessaloni′ca, where there was a synagogue of the Jews. 2 And Paul went in, as was his custom, and for three weeks he argued with them from the scriptures, 3 explaining and proving that it was necessary for the Christ to suffer and to rise from the dead, and saying, “This Jesus, whom I proclaim to you, is the Christ.” 4 And some of them were persuaded, and joined Paul and Silas; as did a great many of the devout Greeks (who practiced Judaism) and not a few of the leading women (Hebrews/Jews).
5 But the Jews (who practiced Judaism) were jealous, and taking some wicked fellows of the rabble, they gathered a crowd, set the city in an uproar, and attacked the house of Jason (because the Christians were meeting in the house of Jason on this first day of the week, Sunday; the Jews definitely did not attack these Christians on the Jews' Saturday Sabbath Day!), seeking to bring them out to the people. 6 And when they could not find them (Paul and Silas), they dragged Jason and some of the brethren (Christians) before the city authorities, crying, “These men who have turned the world upside down have come here also, 7 and Jason has received them; and they are all acting against the decrees of Caesar, saying that there is another king, Jesus.” 8 And the people and the city authorities were disturbed when they heard this. 9 And when they had taken security from Jason and the rest, they let them go.
10 The brethren immediately sent Paul and Silas away by night to Beroe′a; and when they arrived they went into the Jewish synagogue. 11 Now these Jews were more noble than those in Thessaloni′ca, for they received the word with all eagerness, examining the scriptures daily to see if these things were so. 12 Many of them therefore believed, with not a few Greek women (devout converts to Judaism) of high standing as well as men. rsv
Acts 18:4
4 And he argued in the synagogue every sabbath, and persuaded Jews (who practiced Judaism) and Greeks (devout converts to Judaism). rsv
A Greek convert to Judaism was still called a Greek. He never was called a Jew.
These Greeks were devout converts to Judaism. Paul preached to them in the synagogues on Saturday because that is where these Greeks who believed in Judaism were gathered on Saturdays. It is a better plan to go to preach to a group of people who are gathered in just one location than to stop people on the street to preach to each one individually.
The Saturday Sabbath was set aside only for the Israelites/Jews and the converts to Judaism under the Law of Moses/Old Covenant.
Non-Jews or non-converts to Judaism were never required to keep the Sabbath of the Jews during the duration of the Old Covenant.
Christians, whatever their nationality, are under the New Covenant Law of Christ and so they are not ever required to be under by the Law of Moses/Old Covenant Sabbath laws. The Law of Moses/Old Covenant Sabbath laws ended when Jesus died on the cross. Jesus Christ's New Covenant began when Jesus died on the cross. The Cross of Christ ends/finishes one covenant and begins a new covenant.
Romans 10:4
For Christ is the end of the law (of Moses/Old Covenant),
that every one who has faith may be justified (by the Law of Christ/New Covenant). rsv
Acts 15:12-21
12 And all the assembly kept silence; and they listened to Barnabas and Paul as they related what signs and wonders God had done through them among the Gentiles. 13 After they finished speaking, James replied, “Brethren, listen to me. 14 Simeon has related how God first visited the Gentiles (the Gentile Abraham), to take out of them a people for his name. 15 And with this the words of the prophets agree, as it is written,
16 ‘After this I will return,
and I will rebuild the dwelling of David, which has fallen;
I will rebuild its ruins,
and I will set it up,
17 that the rest of men may seek the Lord,
and all the Gentiles who are called by my name,
18 says the Lord, who has made these things known from of old.’
19 Therefore my judgment is that we should not trouble those of the Gentiles who turn to God, 20 but should write to them to abstain from the pollutions of idols and from unchastity and from what is strangled and from blood. 21 For from early generations Moses has had in every city those who preach him, for he is read every sabbath in the synagogues.”rsv
James is speaking about the
Gentile Abraham as the father of the
early generations of the Israelites/Jews. Since the time that synagogues were invented, all the adherents to Judaism listened to readings from the five books of Moses in their synagogues every Sabbath.
There were absolutely no Gentile Christians who lived in these early generations of Judaism. Christianity was still in its very first generation at the time James made this statement.