There is a word, Ἰουδαϊσμός, which refers to Judaism, the religion of the Jews. But so far as we have found, in Biblical literature it is only found in 2 Maccabees, 2:21 and 14:38. The same word Ἰουδαϊσμός appears in Hellenistic Greek literature, as Greek writers described the customs and religion of Judaea under that blanket term.
From the King James Apocrypha:
2 Maccabees 2:21 And the manifest signs that came from heaven unto those that behaved themselves manfully to their honour for Judaism: so that, being but a few, they overcame the whole country, and chased barbarous multitudes...
2 Maccabees 14:38 For in the former times, when they mingled not themselves with the Gentiles, he had been accused of Judaism, and did boldly jeopard his body and life with all vehemency for the religion of the Jews.
But this "Professor Gray" character, a Jew himself, is correct that Jesus is not a Jew, but only a Judaean, and that "Judaism" was not used internally by Jews. He is also correct, that the religion of the temple in Jerusalem is not properly Judaism as it developed subsequent to 70 AD. For example, there was no chicken-twirling in ancient Israel!