Lithium batteries (the usual kind of "watch battery") do not contain sulfuric acid. They contain lithium salts (which are mildly acidic) dissolved in an oil-like solvent. They can, however, leak when they become old or exhausted, though you can easily see this coming out of the seam between the two halves of the battery.
Even if it did leak, the corrosion products are not volatile, so should only cause problems on coins the battery actually touches. It shouldn't cause problems throughout the entire box.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis