"Greek" as a Label for a Gentile Convert to Judaism
Gentile, ἔθνος, is different than Greek, Ἕλλην, and the passage should be approached accordingly. There is a nuance which requires Paul to make a distinction between the two, at least in the particular arguments being made. In this case it appears he intends "Greek" to identify someone who is Jewish but not by birth, or κατὰ σάρκα, according to the flesh (cf. Romans 1:3; 4:1; 9:3, 5).
Paul states all people will be judged by the Law which they know. However, he distinguishes between those who are Jewish by birth and those who became Jewish by choice:
9 There will be tribulation and distress for every human being who does evil, the Jew first and also the Greek, 10 but glory and honor and peace for everyone who does good, the Jew first and also the Greek. 11 For God shows no partiality. 12 For all who have sinned without the law will also perish without the law, and all who have sinned under the law will be judged by the law. 13 For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified. 14 For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. 15 They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them 16 on that day when, according to my gospel, God judges the secrets of men by Christ Jesus. (Romans 2)
From one perspective of Judaism people are born as either Jewish or Gentile. However, a distinction strictly along the lines of ancestry fails to accurately label Gentiles who convert to Judaism as adults. At the same time calling a Gentile convert a Jew obscures a distinction from someone who is Jewish by birth. For his arguments Paul, needs to preserve that distinction and chooses to do so by labeling all Gentiles who are converts to Judaism as "Greek," Ἕλλην.
Why all Gentiles Converts are Labeled as Ἕλλην
Conversion to Judaism was not limited to Greeks. Undoubtedly the Jewish community in Rome would have Gentiles who technically were not Greeks. The justification for labelling any Gentile convert as "Greek" is the language by which they came to know and accept the Mosaic Law:
For from ancient generations Moses has had in every city those who proclaim him, for he is read every Sabbath in the synagogues. (Acts 15:21 ESV)
Practically speaking, the reading in the synagogue would have been from the LXX as a Hebrew reading would have been unintelligible to Gentiles. In other words, the manner in which a Gentile was taught the Jewish religion was using the Greek language. The service in the synagogue would either be given collectively in Hebrew and then Greek, or conducted separately based on language.
The Greek who practiced Judaism could be identified as Jewish for legal reasons
2and yet ancestrally they had no tribal affiliation. In some ways they could be considered as a "lesser" type of Jew since they lacked the promise of a specific earthly inheritance in Israel. They would always be foreigners, unless they married someone who was Jewish by birthright. Even that act, in the strictest sense would not change their status, since one who was Jewish was not to marry outside the descendants of Abraham.
It is essential to Paul's argument of justification by faith that he preserves a distinction within Judaism between the Gentile convert and one who is Jewish by birth. True, both are Jewish in the sense of religion, but the Gentile convert is Jewish by choice, and, one could rebut Paul by claiming that choice was by faith. Hence, to block the counter argument, Paul recognizes and preserves the existing distinction within Judaism by labeling all Gentiles who consider themselves to be Jewish as "Greek." They too will be judged according the Law; a judgment which comes after those who are Jewish by birth.
In fact, the previous actions of the Jewish community to witness to the Gentiles in order to bring the knowledge of God to a pagan world, is based on the principle of justification by faith. Paul's point is that faith was always misplaced in the Law's ability to bring salvation.
1. The existence of this type of division within Judaism is also evident from the early Christian Church in Jerusalem: Now in these days when the disciples were increasing in number, a complaint by the Hellenists arose against the Hebrews because their widows were being neglected in the daily distribution (Acts 6:1). This early Church was composed entirely from those who were Jewish. Therefore, the division within the Judaism, or the synagogues was carried into the Christian Church. The Greek speaking Christians are called Ἑλληνιστῶν, a term which Paul avoids, likely to prevent confusion with those who reject the Gospel. So then, a Gentile who had converted to Judaism who became a Christian was a Ἑλληνιστῶν and one who remained Jewish would be a Ἕλλην.
2. Judaism was recognized as a legal religion which allowed certain privileges, a condition which would be more pronounced in Rome than in the provinces.