You're misinterpreting the meaning of the order status named "Processing". It does NOT mean that the *payment* is still processing. The word "Processing" is merely the word configured to reflect that payment has been received and now the ORDER is in a "Processing" status which means it's waiting on your shipping department to "process" the order for shipment and get it out the door. If you don't like that word, you can rename it in Admin->Localization->Order Status

A payment module connected to a live gateway SHOULD be set to "Processing" as the status used to apply to orders paid for with that module. That's why it's the default setting for the module when it's first installed/enabled.