Assuming you are trying to create the geodatabase by using ArcCatalog please remember that ArcCatalog is a 32 bit application. So you need the corresponding 32 bit Oracle client independent if your database is 32 or 64 bit.

As Midavalo, lele3p and Vince already mentioned the Oracle database software has to be 64 bit. The use of 32 bit databse is obviously not only ‘not supported’ it seems to be explicit excluded.

At the client side, assuming you are trying to create the geodatabase by using ArcCatalog please remember that ArcCatalog is a 32 bit application. So you need the corresponding 32 bit Oracle client independent if your database is 64 bit. When you are using the 64 bit Background Geoprocessing you’ll need the 64 bit Oracle client instead. Geoprocessing background 64 bit