Has a reduction in car use ever had an effect on the ozone layer?
The sequence of events you describe has never happened, for several reasons.


As Sabre Tooth mentions in the comments, vehicle emissions have a negligible effect on stratospheric ozone. (Note that while vehicle emissions can lead to ozone production at ground level, the ozone layer is several kilometres above the Earth's surface and isn't really affected by variation in ground-level ozone. The EPA has published a guide entitled Ozone: good up high, bad nearby which explains the distinction in more detail.)
Even the most dramatic attempt to reduce driving in the USA, the National Maximum Speed Law of 1974, only produced a 1% (or less) reduction in fuel consumption; we can assume that reductions in emissions would have been of similar magnitude.
Even in 1974, there were a lot of vehicles outside the USA, diluting the global impact of any changes in driving habits within the USA.


However, to answer a slightly broader question, it does seem that the ozone layer is capable of repairing itself when emissions of ozone-depleting substances are reduced. The Montreal protocol was enacted in 1989:


  The Montreal Protocol on Substances that Deplete the Ozone Layer was designed to reduce the production and consumption of ozone depleting substances in order to reduce their abundance in the atmosphere, and thereby protect the earth's fragile ozone Layer. The original Montreal Protocol was agreed on 16 September 1987 and entered into force on 1 January 1989.


A report released in 2014 by the World Meteorological Organization concluded:


  Actions taken under the Montreal Protocol have led to decreases in the atmospheric abundance of controlled ozone-depleting substances (ODSs), and are enabling the return of the ozone layer toward 1980 levels.


They estimate that total recovery will occur "before midcentury in midlatitudes and the Arctic, and somewhat later for the Antarctic ozone hole".