Matt wrote:
My water softener has leaked hard water very slowly for some years.

That made the floor damp and caused a buildup of lime on the concrete
floor. The lime is probably not more than quarter millimeter thick but
is unsightly.

My impulse is to use Lime Away, but I expect that that would attack the
cement in the floor.

Is there any good way to get rid of these lime stains?

Phosphoric acid.