If a character gets an 'acid bath' and then healed right away, would any previous scaring or tattoos (non-magical) be gone or does the skin heal back to the way it was before?
Scars are fibrous connective tissue that forms over areas where skin cannot regrow or simply can't regrow fast enough to seal up the wound in good time. A lot of small scars disappear over time because the skin eventually regrows to replace the connective tissue. Healing magic would repair the wound rapidly, so scar tissue wouldn't be necessary.
I'd say all the damage that is healed would be fresh, new, pink skin. If the scar is in an area that doesn't need healing, I wouldn't have it be removed by a simple cure. That's what regeneration spells are for.