Q. If, as you say, the law was given only to Israel, why does Romans 3:19 imply that all the world is under the law and judged guilty by it?
A. The first Christians in Rome appear to have been Jewish. It is quite possible that they were among the Jews from various parts of the empire who heard Peter speak on the day of Pentecost as recorded in Acts 2. When they returned to Rome, they brought Christianity with them and spread it to other Jews there. Thus, the Christian community in Rome had a distinctly Jewish character. Apparently some Gentiles also became Christians in Rome, but they were a minority. As we know from Acts and Paul's epistles, some Jewish Christians not only kept many Jewish traditions but they also had a difficult time understanding the fulfilling and ending of the law. Sometime around A.D. 49, the Roman emperor, Claudius, issued an edict expelling Jews from the city of Rome. The edict would have included even those Jews who had converted to Christianity. But it did not include the Gentile Christians, who stayed behind in Rome. This expulsion of the Jews is mentioned in Acts 18:2. Some years later, the edict was either withdrawn or allowed to lapse with the death of Claudius in A.D. 54.
When the Jewish Christians returned to Rome, they were dismayed to find that the Christian community had gone on without them and was now decidedly Gentile in character. The Gentile Christians, on the other hand, understanding the ending of the law and not having been Jews, did not want a return to Jewish ways. This intolerance between the two groups resulted in friction. This tension between the Jewish and Gentile Roman Christians is at least one of the reasons Paul wrote his letter to the Romans, and this background should be kept in mind when reading the epistle. It will certainly help us understand the answer to this question about Romans 3:19.
In the passages leading up to Romans 3:19, Paul is trying to show both sides in the dispute—the Christian Jews and the Christian Gentiles—that neither party is superior. To do this, he is developing (among other things) two argument threads: one showing how the Gentiles are under judgment and the other showing how the Jews are equally under judgment. By the time we get to Romans 3:19, Paul has already shown how the Gentiles are under judgment "without law" (see chapter 2). All Paul now has to do to show "all the world" to be under judgment is to show how the Jews are guilty.
The first part of Romans 3:19—"Now we know that what things soever the law saith, it saith to them who are under the law"—is an argument in Paul's thread showing the judgment of the Jews. The Jews are under the law, it speaks to them, and they are condemned by it. Because of this, "every mouth may be stopped." I think that Paul here has the Jewish mouths particularly in view. The Jewish Christians thought they were superior to their Gentile brothers in Rome and they were mouthing off against them. But Paul shows that, contrary to what the Jews were thinking, the law does not make the Jews superior, it condemns them.
Thus, it is in this way that "all the world"—including the Jews—may become guilty before God. Because of their law-keeping, the Jews thought they were an exception to this guilt. But, in fact, the law shows their guilt.
So, as a direct answer to the specific question, the law given to the Jews causes the whole world to become guilty before God because Paul has already shown that the Gentiles who "have sinned without law shall also perish without law" (Romans 2:12). So Romans 2 shows that the Gentiles—who are a portion of the world—are guilty. But this is not yet "all the world"; the Jews are missing. So, with this knowledge, we enter into Romans 3 where Paul shows that the law makes the Jews guilty. Now we see how "all the world may be under judgment before God." It means everybody, including Jews. To reiterate, in Romans 2, Paul shows how the Gentiles come under judgment without the law; and in Romans 3, he shows how the Jews come under judgment under the law: thus all the world comes under judgment. In this way, Paul shows to the Jewish Christians and to the Gentile Christians that neither should consider themselves superior.
Where is boasting then? It is excluded.
By what law? of works? Nay: but by the law of faith. Therefore we
conclude that a man is justified by faith without the deeds of the law.
Is he the God of the Jews only? is he not also of the Gentiles? Yes, of
the Gentiles also.