Do I just leave it? PC case had rubber legs but it was stuck to the floor and when I tried to flip it, part of the wooden floor came off with it.
What came off is the finish (urethane, polyurethane, or varnish), not the wood.
If left alone, it will absorb dirt, moisture and darken.
I would try a wipe on polyurethane and apply it with a q-tip. Your finish looks like gloss, so I would try that level of sheen. You can always dull it a bit with fine steel wool.
You should try to remove the rubber stain first. That can be done with a cleaner, such as Simple Green or Fantastik type cleaners.