Can I wire a box with two 120-volt duplex outlets each wired to one leg of a 240-volt 3-wire plug?
Your question should be "can I use ground as neutral" since your 6-20 has only 2 hots and ground, no neutral. It is a against code and potentially dangerous to use a ground in place of a neutral.

You need to pull a separate neutral wire back to the panel or sub panel. If you want to keep 240V available, install an L14-20 outlet and use an adapter off of that.