Can I use 6/3 NM-B cable for a 120v 20 amp circuit?
Sure, you're always allowed to use oversize wire.  You are designating the circuit 20A, you could use anything 12 gauge and up.  This is absolutely fine, there is nothing wrong with this, in fact it's a good idea if it's a long run and copper is no object. 

You just have to contend with physical issues: the outlet (and possibly the breaker) is not listed to accept 6 gauge wire.  Just pigtail the outlet with 12ga wire and use a connector listed to join 12ga wire to 6ga wire such as Ideal 76B (their red wire-nut).

I don't think you'll have box fill issues for one outlet, but you'll have to start thinking about that if you branch off any more wires.  Outdoor boxes are often larger than indoor ones.  If you're not using the existing box, you'll want to use at least a 2-gang box that's deep - most outdoor boxes should suffice. Thanks Speedy.