yeah, when the gentiles separated from jews, they followed paul and barnabas and the verse said "the Gentiles begged that these words might be preached to them the next Sabbath".
this would have been a great teaching moment for paul to tell the gentiles that they don't meet on the sabbath because it was done away with, and now the christian is to meet on the "first day of the week".
How well do you think your advice would have been received by the Jews in charge of the synagogues if Paul and Barnabas had done as you suggested?
The Gentiles who were in the synagogues practiced Judaism or else they were there to learn about Judaism in order to convert to Judaism. Please note that there is no Scriptural record that any other Jewish Christians or Gentile Christians accompanied Paul and Barnabas into the synagogues. Barnabas accompanied Paul into the synagogues.
Acts 13:43 And when the meeting of the synagogue broke up, many Jews and devout converts to Judaism followed Paul and Barnabas, who spoke to them and urged them to continue in the grace of God. rsv
Matthew 10:17
Beware of men; for they will deliver you up to councils, and flog you in their synagogues, rsv
Acts 13:50
But the Jews incited the devout women of high standing and the leading men of the city, and stirred up persecution against Paul and Barnabas, and drove them out of their district.
It seems reasonable to believe that "when the meeting of the synagogue broke up", Paul and Barnabas did indeed 'invite" the "many Jews and devout converts to Judaism who had followed Paul and Barnabas" out of the synagogue after the meeting to join them on the first day of the week to learn more about Jesus Christ so that they could then "continue in the grace of God".
It also seems reasonable to believe that when the Jewish leaders of the synagogue learned that Paul and Barnabas had "invited" their fellow Jews from their own synagogue "to join them on the first day of the week to learn more about Jesus Christ", that they then drove Paul and Barnabas out of their district.
paul didn't say that! in fact, paul told them nothing like what you said,
"The Christian Gentiles met together for worship on the first day of the week."
remember paul said to peter, "why do you compel Gentiles to live as Jews?"
if paul stood up to peter, surely he would have told the gentiles what all of the sabbath haters on this forum say about the sabbath!
again, he didn't!
they were gathered together to eat a meal. that's what the verse says!
can't make it say something it's not saying! 
The correct translations of the Bible say that it was Cephas who Paul rebuked, not Peter. Also, the early church historian Eusebius said that it was Cephas, one of the 70/72 disciples, who was rebuked by Paul. I'll stick with these sources.
According to the law of Moses, it was not lawful for a Jew to eat with Gentiles. But the Jews who converted to Christianity were supposed to eat with Gentiles. They were to make no distinctions between themselves and the Gentile converts.
Galatians 2:11-16
But when Cephas came to Antioch I opposed him to his face, because he stood condemned. 12 For before certain men came from James, he ate with the Gentiles; but when they came he drew back and separated himself, fearing the circumcision party. 13 And with him the rest of the Jews acted insincerely, so that even Barnabas was carried away by their insincerity. 14 But when I saw that they were not straightforward about the truth of the gospel, I said to Cephas before them all, “If you, though a Jew, live like a Gentile and not like a Jew, how can you compel the Gentiles to live like Jews?” 15 We ourselves, who are Jews by birth and not Gentile sinners, 16 yet who know that a man is not justified by works of the law but through faith in Jesus Christ, even we have believed in Christ Jesus, in order to be justified by faith in Christ, and not by works of the law, because by works of the law shall no one be justified. rsv
So, why do you think that Paul rebuked Cephas? Cephas was a natural-born Jew who converted to Christianity. Paul rebuked Cephas because Cephas lived like a Gentile and ate with the Gentiles as he was supposed to do unless there were disobedient Christian Jews of the circumcision party present. When the troublesome Jewish Christians came to visit, Cephas compelled the Gentile Christians to separate themselves from the Jewish Christians whenever they ate. Christians are
not allowed to follow the law of Moses and separate themselves from the Gentiles while eating. The law of Moses is not valid for Christians.
Sunday Worship
Christians not only ate together, they also worshiped together and partook of the
breaking of the bread that was commanded by Jesus Christ.
Eating meals together and
the breaking of the bread during their worship services are
two different things.
1 Corinthians 11:24
Christians learned from their leaders where to meet for Sunday worship. They did not need a letter written to them in order to know where and what time they were to meet on the first day of the week for their worship service. The Scriptures are supplements to the apostles' and disciples' teachings. The Scriptures do not take the place of their teachings and they do not contain all of the apostles' and disciples' teachings.
2 Thessalonians 2:15
So then, brethren, stand firm and
hold to the traditions which you were taught by us,
either by word of mouth or by letter. rsv
John 21:25
But there are also many other things which Jesus did; were every one of them to be written, I suppose that the world itself could not contain the books that would be written. rsv
Christians do not ever convert to Judaism. If they do, they have denied the Christian faith.
Galatians 5:4
You are severed from Christ, you who would be justified by the law; you have fallen away from grace. rsv
