MAURA (2) f Scottish, Irish, English Pronounced: MOR-a Possibly from Gaelic mór meaning "great". This was the name of an obscure 5th-century Irish or Scottish martyr. This name is also used as an Anglicized form of MÁIRE (MAIRE).
Click the links provided to find out the meanings of Maurus / Maire. As you can see, Maura as derived from the second root is indeed an Irish form of Mary.
Sorry if I seemed harsh, I'm just getting very impatient with people who don't see the search box.