As with so many other things, it depends:
Receptacle at the end of the circuit
If this receptacle is the last device in the circuit - i.e., one cable comes in to power the receptacle but no other cables are in the box, then you can:
- Disconnect the cable from the receptacle.
- Disconnect the cable from the source (which may be elsewhere in the bathroom or could be in another room or could be all the way back at the panel if this receptacle was the only device on the circuit). You MUST get that cable OUT of the other box or panel so that it won't get somehow reused with the other end hanging in the wall. Ideally, you would remove the cable altogether, but if it is stapled to the wall frame somewhere then that will not be practical. Just make sure it is out of the box.
- Remove the receptacle and box and patch the wall.
Receptacle in the middle of the circuit
If this receptacle is in the middle of the circuit - i.e., there are two cables coming into the box, one to power the receptacle and another going on to someplace else (more receptacles, lighting, etc.), then you can't remove the box. You have two options:
- Move the receptacle and box RECOMMENDED
Move the box to the other side of the wall. Note that depending on how the existing box is connected, it may be easier to rip out the old one and put in a new box for the other side.
Splice the hot, neutral and ground wires between the "in" and "out" cables, using an acceptable method. Note however that ordinary wire nuts are absolutely not acceptable inside an inaccessible portion of the wall. You can read more about it in this question. It is not clear to me whether this is legally an option in this specific situation where you are not repairing a cable (e.g., as might happen if someone managed to drill or nail through the wall into an existing cable which would then need to be repaired without putting in a box) but rather making some changes where, as you already stated, the other side of the wall is accessible.
As should go without saying, but I'll say it anyway, even if your old receptacle was not GFCI protected, any new bathroom receptacles absolutely should be. This is both a code requirement and common sense. It is quite possible that the old receptacle was on a GFCI breaker and the new receptacle "from nearby wiring" is not, in which case you would need to add a GFCI receptacle combination device or change the breaker. The opposite is also possible - the existing receptacle might be a GFCI combo but the new receptacle might be on a GFCI breaker - bottom line is that the new receptacle should have exactly one GFCI protection in place. More than one doesn't make you safer but does make it harder to troubleshoot problems.
There are code requirements related to having a dedicated, or shared with only certain other things, receptacle circuit in each bathroom. I don't know how that relates to renovation vs. new work or what the requirements are exactly. One of the pros will chime in, I'm sure, if that is relevant to your situation.