You can test with a vitamin C tablet or, better, some Iron Out (laundry section) held in a wet rag against some of the spots. If either of those lighten the spots, it's iron. Thousands of rusty brown spots sound like someone was grinding or cutting something -- maybe rebar for the deck? -- and didn't watch where the filings went. Do NOT let the contractor take a pass on this one; they may not be possible to remove completely.

Ben