Hi,
Support for the File Geodatabase format is only available when using the LocalServer component of the ArcGIS Runtime SDKs. This supports creation/read/write and traditional check-out/check-in replication workflows. Direct read/write from the ArcGIS Runtime itself is not currently on the roadmap.
The new Mobile geodatabase (*.geodatabase) is the main workflow for ArcGIS Runtime (and support is coming to ArcGIS Desktop/Pro too). Edits can be made to the mobile geodatabase whilst your app/device is offline but the geodatabase must have been extracted initially from a sync-enabled feature service to be able to make and save/synchronize edits. Once your app/device is back online you can synchronize any edits based on your specific workflow (bi-directional, upload-only, download-only, etc).
Cheers
Mike