Some people that believe in keeping aspects of Mosaic law object to the term "Judaizer" or "Judaizing" saying it a made up term, and that its only used by Law (Torah) breakers.
However, the term Judazing is used in the New Testament, in fact, by St Paul himself
But when I saw that they were not straight forward about the truth of the gospel, I said to Cephas in the presence of all, “If you, being a Jew, live like the Gentiles and not like the Jews, how is it that you compel the Gentiles to live like Jews [Ioudaizein]?-Galatians 2:14
Ἀλλ' ὅτε εἶδον ὅτι οὐκ ὀρθοποδοῦσιν πρὸς τὴν ἀλήθειαν τοῦ εὐαγγελίου, εἶπον, τῷ Πέτρῳ ἔμπροσθεν πὰντων, Εἰ σύ, Ἰουδαῖος ὑπὰρχων, ἐθνικῶς ζῇς καὶ οὐκ Ἰουδαϊκῶς, τί τά ἔθνη ἀναγκὰζεις Ἰουδαΐζειν;--Galatians 2:14
As we can see, St Paul used the word with the meaning "live like a Jew." Sabbath keeping is never pushed by Apostles in the New Testament, it was only obligatory until Christ's death, and was tolerated afterwards in the New Testament as part of the transition period for the Jews accustomed to keeping the 7th day sabbath.