Jews or Judeans were descendants of Abraham's great-grandson Judah.
So, the short answer is no, he couldn't have been !
Literally the term Jew means : Of Judah. A person belonging to the tribe of Judah.
The name is not used in the Bible account prior to the fall of the ten-tribe kingdom of Israel where the southern kingdom was called Judah, and the people were called sons of Judah or the tribe of the sons of Judah.
The term Jew first appears in the Hebrew Bible in Jeremiah’s day, at Jeremiah 34:9, and also in Esther concerning Mordecai the Jew, and in the postcaptivity prophecy of Zechariah 8:23.
Later all Israelites who held to the tribe of Judah because of their exclusive worship to Jehovah were also called Jews or Judeans, and the name is so used in the Christian Greek Scriptures.
This elicits another question: Who is a jew today ?
It seems it depend on who you ask.
After the exile (from Babylon) the name was applied to any Israelites returning to Israel (Ezr 4:12; 6:7; Ne 1:2; 5:17) and, finally, to all Hebrews throughout the world, to distinguish them from the Gentile nations. (Es 3:6; 9:20)
Gentiles who accepted the Jewish faith and became circumcised proselytes declared themselves Jews as well.
The Israeli Supreme Court (1970). held,that anyone was a Jew who claimed to be a Jew, even though he was an atheist and not born of a Jewish mother.
However, Orthodox Jewry maintains that Jewish religion and nationality are inseparable.
Tthe Knesset, (Israeli parliament), declared that the definition of Jewishness in the Halakah, be henceforth declared the only legal one. (The Halakah defines a Jew as one born of a Jewish mother or a convert to Judaism). This was made law on March 10, 1970.
Interestingly, The apostle Paul, in arguing that the Jews were mistaken in their pride of fleshly descent and in relying on the works of the Law to find favor with God, said: “For he is not a Jew who is one on the outside, nor is circumcision that which is on the outside upon the flesh. But he is a Jew who is one on the inside, and his circumcision is that of the heart by spirit, and not by a written code. The praise of that one comes, not from men, but from God.” (Ro 2:28,29)
This argument parallels his reasoning in Romans chapter 4, that the true seed of Abraham are those with the faith of Abraham not fleshly lineage.
Does it matter ?
Maybe Christ's words puts some perspective on it :“ there is neither Greek nor Jew, circumcision nor uncircumcision, foreigner, Scyth′ian, slave, freeman, but Christ is all things in all”.(Col 3”10)