Hello,
Potentially you could, but it would require you to create a new Cape PCB (i.e. to fit the Raspberry Pi rather than Beaglebone Black) and also adjust software to fit the new operating system.
It is unclear to me why you wish to do this as it would not save any space – just replace one microcontroller with another.
If you want to run the existing Beaglebone Black _without_ connecting it directly to a PC then this is possible – you just have to set your network up to connected to it via ethernet. I believe others have done this in the past but I have not as we just connect a bunch of them to PC via long USB cables 🙂