The Arduino would have to be the USB master. Currently, it's not even a slave, which is a simpler circuit. It's just wiggling UART bits to be a serial device.
The Arduino would have to know the usbstorage protocol to talk to the thumbdrive. I don't know if that's just like SD or not.
The Arduino would have to treat the flash capacity as a vanilla block device, or support a filesystem. There are software implementations available to support the FAT12/FAT16 you see on most SD cards and thumbdrives. With these libraries, your limitations are short filenames and no subdirectories.
I was thinking that writing to USB "thumb" drive wouldn't be appreciably different than writing to an SD card. What I need to do is go from a serial connection on a piece of test gear and store the data on something that's harder to lose (physically) than an SD/MMC card -- the test equipment is in the field (no place I'd want to take a laptop), and it would be a WHOLE lot easier swapping out solid-state memory than hauling the thing back to the shop to download to a PC.