Can 12-2 Romex run close to the floor and the back up to the box?
Two concerns that I know of:

Water

If the wire is at least a few inches off the floor then this is a non-issue. But if it is literally on the floor then water - whether deliberate from washing the floor or accidental from an overflow - could get into the cable and cause problems because standard NM cable is not rated for wet areas.

Physical Protection

As with any other cable up to (I think) 8 feet above the floor, cable needs to be protected from damage. This can be conduit or raceway (wiremold) or it can be wood screwed into place around the cable to protect it from accidental damage.
As noted in a comment, conduit + THWN wire would take care of both potential problems.