Does Reverse Osmosis remove elemental ions from water?
The short answer is pressure. What you've described above is basically just osmosis, and without applied pressure to overcome the osmotic pressure of the system, reverse osmosis doesn't happen.  

This Wikipedia article gives a good description of the concept, and the difference between filtration, osmosis and reverse osmosis:  


  Osmosis is a natural process. When two solutions with different
  concentrations of a solute are separated by a semipermeable membrane,
  the solvent has a tendency to move from low to high solute
  concentrations for chemical potential equilibration.
  
  Formally, reverse osmosis is the process of forcing a solvent from a
  region of high solute concentration through a semipermeable membrane
  to a region of low solute concentration by applying a pressure in
  excess of the osmotic pressure. The largest and most important
  application of reverse osmosis is the separation of pure water from
  seawater and brackish waters; seawater or brackish water is
  pressurized against one surface of the membrane, causing transport of
  salt-depleted water across the membrane and emergence of potable
  drinking water from the low-pressure side.
  
  The membranes used for reverse osmosis have a dense layer in the
  polymer matrix—either the skin of an asymmetric membrane or an
  interfacially polymerized layer within a thin-film-composite
  membrane—where the separation occurs. In most cases, the membrane is
  designed to allow only water to pass through this dense layer, while
  preventing the passage of solutes (such as salt ions). This process
  requires that a high pressure be exerted on the high concentration
  side of the membrane, usually 2–17 bar (30–250 psi) for fresh and
  brackish water, and 40–82 bar (600–1200 psi) for seawater, which has
  around 27 bar (390 psi)[6] natural osmotic pressure that must be
  overcome.