Will increased level of ocean acidity lead to more frequent acid rain?
More likely, it works the other way: increased acid-forming compounds in the atmosphere are the cause of more ocean acidification.
Pio et al. [1] report measurements of sulfur and nitrogen compounds in air masses that they traced to either continental or oceanic origins.  They report that the lowest levels of these compounds were from ocean-origin air masses, indicating that the ocean is absorbing acid-rain causing pollutants such as sulfur dioxide.  Even so, levels of these compounds over the oceans has increased in recent years, likely from increased pollution on land.
In this answer it is noted that increasing acidity can alter the salt composition of the ocean, producing more acidic species (such as $\text{H}_2\text{PO}_4^-$ ions).  However, these tend to remain dissolved in the water rather than evaporate into the atmosphere.
Reference
1.
Casimiro A. Pio, Mário A.Cerqueira, Luis M.Castro, Maria L.Salgueiro, &quot;Sulphur and nitrogen compounds in variable marine/continental air masses at the southwest European coast&quot;, Atmospheric Environment
Volume 30, Issue 18, September 1996, Pages 3115-3127.