Can I just patch this? Or do I need to cut out the drywall around the hole and add new drywall around the outlet? I plan on sealing the rest of the damaged drywall with Gardz and will be tiling over it.
Spackle should work.
For a deep patch like this I would use Ready Patch. It is available in a half pint.
Here is a link that might be useful: Ready Patch
At the big box stores they sell drywall hole patches - square in various sizes. Thin metal, fiberglass tape backing.
Buy one and use tin snips to cut a portion of it off and stick over the crumbled hole. Mud over it.
If you fill it with ready patch you can cover all of it with an oversized wall plate and avoid using multiple coats of compound, sanding and perhaps paint.
The wall finish seems a bit strange. Maybe you plan to refinish all of it anyway.