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Nope, you cannot. Sorry.
I have the same deployment as you. You have to add your new SQL Express instance to your Database Servers in Catalog, and then you just right-click that instance> 'new Geodatabase'.
Let me know if you are having any trouble adding your new instance.
see http://forums.arcgis.com/threads/61313-Failed-to-connect-DB-Bad-login-user
1. Open registry and you will notice that the entry 'ArcSDE_License' is missing:and still didn't work. Same error "Failed to connnect to database. Bad login user. Succes:)))''
'HKEY_LOCAL_MACHINE', 'SYSTEM\CurrentControlSet\Services\MSSQL$sqlexpress'
2. Now , manually create an 'ArcSDE_License' entry as New String Value. You will get the needed value
for this entry with steps below:
a. Use the following URL to generate ECP file: https://customers.esri.com/index.cfm...Authorizations
b. Modify and copy string from ECP file into the ArcSDE_License entry.
3. Relaunch ArcCatalog and one will be able to create a new geodatabase