Is it accurate to call Abraham a Jew?

It is common to associate Abraham as "father of the faith" and the first Jew. However, since Abram's family came from the land of Ur, isn't it more accurate to say that he wasn't a Jew but - in reality - a Babylonian that **became** a Jew because of his faith in God? This is open for answer by all denominations. I would like to hear their thoughts on the matter.