No, he was not. The word "Jew," a nickname for the Israelite tribe of Judah, originated in the thirteenth century. It is first used in the Bible to identify the "men of Judah" in II Kings 16:6 (King James Version), a thousand years after the time of Abraham.
The tribe of Judah descended, as the name indicates, from the man named Judah, one of the twelve sons of the patriarch Jacob (Genesis 35:23). Jacob was a son of Isaac (Genesis 25:26), and Isaac, in turn, was a son of Abraham (Genesis 21:3). We can see, then, that Judah, the progenitor of the Jews, was a great-grandson of Abraham. Thus, Abraham was not a Jew, but an ancestor of the Jews. Abraham is properly referred to as a "Hebrew," a descendent of Eber, a great-great-grandson of Noah (Genesis 11:10-26).