Your pool water chemistry should have been monitored closely for a 2 week period and especially the first 3 days. Total alkalinity, calcium hardness and pH. The water temperature plays a part in the curing process and your test results. It's a good idea to test your fill water also as a starting point.

A sequestering agent should have been added upon filling to prevent any issues from iron, calcium or any other minerals from falling out of a solution form and becoming a solid. These solids hang onto surfaces.

NO auto pool cleaners should be used for a minimum of 2 weeks. Initial vacuuming(s) with concrete, wheeled vac heads is no good. A vinyl vacuum head with brushes is the best.
The pool should have brushed twice a day (or more if you have the time) for at least 7 days.

It sounds like the white spots you are seeing is calcium. Most plaster contractors will calcium flake to aid in speeding up setting time. Sometimes too much is added and if the water chemistry is not addressed immediately upon filling, the end result could be what you have.

There is a process to drop the water chemistry to an acid state to expose the finish and help remove calcium stains. The downside of this is that it can sometimes create a rougher finish as the acidic water etches the finish.

It's too bad your contractor did not inform you of the proper start up procedure or at least steer you in the right direction.

There are some products from Jack's Magic or McGrayel Water Technologies that may work.

Have a water sample tested and the information on hand if you call either of these companies as they will want to know test results so they can hopefully resolve your issue.