You could, to EcoRI it doesn't make much of a difference what vector its cut site is on. I don't recommend it for this application though. It looks like both vectors encode for ampicillin resistance so you might have to screen a lot of colonies doing it this way. You can't even control against backbone self ligation because adding a dephosphorylating enzyme would inactivate all fragments present, including your insert.

It's probably better to digest each plasmid separately, gel extract the insert away from the pGEM vector, dephosphorylate the pProExHT vector after the restriction digest, then figure out what insert orientation each clone is in. Or you could just add two convenient restriction sites by PCR, that's the less difficult way.