PS2 can only run PS1 games (close to) natively from optical discs. The system that switches the PS2 over to PS1 mode can only be triggered physically by the drive, and once in PS1 mode the console can no longer access most of the PS2 hardware, including any storage attached to the network adapter.

There is a way to “convert” PS1 games to run from internal storage (POPStation), but it actually just wraps the PS1 image in a software emulator for PS2.