Were the Gentiles allowed to go into the synagogues?
Gentiles were allowed to come to the temple complex, but they were restricted to remain in the outer court, the court of the Gentiles. The question is whether some similar restriction forbade Gentiles from entering synagogues.
We know that Gentiles did believe in God. One even financed the building of a synagogue in Capernaum.
"Now when He concluded all His sayings in the hearing of the people, He entered Capernaum. And a certain centurion's servant, who was dear to him, was sick and ready to die. So when he heard about Jesus, he sent elders of the Jews to Him, pleading with Him to come and heal his servant. And when they came to Jesus, they begged Him earnestly, saying that the one for whom He should do this was deserving, "for he loves our nation, and has built us a synagogue"" (Luke 7:1-5).
The later part of Acts 13:42 is sometimes declared spurious because it clearly shows that Gentiles were in the synagogue and stayed after the Jews had left. But the reasoning is that Gentiles could not be there. In other words, because some scholars did not believe Gentiles would be in the synagogue; therefore, the phrase must be spurious. The line of reasoning is poor, especially in light of other passages.
"And when the Jews were gone out of the synagogue, the Gentiles besought that these words might be preached to them the next sabbath" (Acts 13:42).
By teaching in the synagogue both Jews and Greeks were converted. The implication is that there were Greeks in the synagogue.
"And it came to pass in Iconium, that they went both together into the synagogue of the Jews, and so spake, that a great multitude both of the Jews and also of the Greeks believed" (Acts 14:1).
"Now when they had passed through Amphipolis and Apollonia, they came to Thessalonica, where there was a synagogue of the Jews. Then Paul, as his custom was, went in to them, and for three Sabbaths reasoned with them from the Scriptures, explaining and demonstrating that the Christ had to suffer and rise again from the dead, and saying, "This Jesus whom I preach to you is the Christ." And some of them were persuaded; and a great multitude of the devout Greeks, and not a few of the leading women, joined Paul and Silas" (Acts 17:1-4).
"And the brethren immediately sent away Paul and Silas by night unto Berea: who coming thither went into the synagogue of the Jews. These were more noble than those in Thessalonica, in that they received the word with all readiness of mind, and searched the scriptures daily, whether those things were so. Therefore many of them believed; also of honourable women which were Greeks, and of men, not a few" (Acts 17:10-12).
"And he reasoned in the synagogue every sabbath, and persuaded the Jews and the Greeks" (Acts 18:4).