I have a wall mount oven/microwave that has a 50 amp breaker servicing it. The microwave was attached to and plugged into the oven. I removed both and only want to replace the microwave. However, I only have the 6 gauge wire stubbed into the cavity. How can I add a 120 outlet for the microwave using the 6 gauge wire? Can I pigtail 14 gauge wire to the 6 gauge and run that to another box and put the outlet in that box?
You don't... not with those wires :(
The problem is the cable you are looking at there is a "Type SE, Style U" cable. The giveaways are the red+black hot conductors, and the scruffiness of the bare wire; in the cable jacket those individual strands are wrapped around the conductors, and they've been twisted together here.
SE style U cable is a funny duck. It's made for service entrance wiring, which is the wiring from the poletop to the meter to the main panel. On the pole side of the main panel, there is no ground (ground is established at the main panel) and so the bare wire is neutral.
The good news is, that was a legal way to install a 120/240V oven circuit in 1973. (well gosh, up til 1990 if someone had old stock of #6 SEU cable to use up). You can't install a new oven circuit like that, but if it was legal when it was installed, you can continue to attach it to new 120/240V ovens and ranges. I'm not a fan though; the Code changed for a reason.
SEU cable of that size was NEVER legal to support any 120V appliances.
A 6AWG "hot" wire cannot be re-marked to use as a neutral. So it is not good enough to re-mark the red wire "white" and re-task the bare mesh "wire" to be safety ground.
You are out of luck. You'll need to pull a new circuit, or arrange it so it can plug in to a kitchen countertop circuit (but it must be plug-in, not permanently installed).
If you pull a new circuit, also replace That 70's Cable with a shiny new 10/3 w/ground. The cables can run together so it's no more work, just material cost which isn't that high compared to labor.