Paul reminds them of their Old Testament, which showed how the Jews themselves all too often stirred up blasphemy among the Gentiles by their own licentious behavior.  When he says that it is written that the name of God is blasphemed among the Gentiles through you, he may perhaps have some reference to Nathan’s accusation of David after his crime of adultery and murder.  There, the prophet told the guilty king that by his act he had given great occasion to the enemies of the Lord to blaspheme.  By just application, we may say that each time a member of the professing church commits some gross outward sin, he yields opportunity to the enemies of the Lord to discount his religion, and revile the name of his God. kjv_romans_2-24

Paul may also have some reference to Ezekiel 36:20, where the prophet spoke of the captive Hebrews profaning His holy name among the heathen by their ill behavior.  The language is also quite reminiscent of Isaiah 52:5, where God complained that His name was continually blasphemed.  Such an accusation, Paul would have them know, fit not just their forefathers, but with they themselves.  Therefore, they had no call to pretend to a righteousness superior to that of the Gentiles, but they themselves stood guilty before God just as the heathen he had described with such dark colors in the first chapter.