#RAW No damage dice => No dice to double
Sucks, but them's the rules. So a critical hit with an unarmed strike does nothing more than 1 damage. The tables in the player's handbook doesn't list a "1d1", just 1 damage. "1" is not a dice roll, it's a number.
#RAI Critical hits are meant to generally do more damage than a normal hit. I'd instantiate a house rule for upping it to a 1d4 or supply 2 damage on a critical hit.
#A Note
5e greatly acknowledges the importance of house rules in D&D culture, so I would not stick with the RAW reading of this. It just is not fun and takes the joy out of a critical hit. Come up with a rule that seems right (doubling to 2 damage, or going to 1d4 damage, or something else) to stick with the RAI.