Scrub it lightly with the sponge to remove the haze.
Remove and replace grout with white marble tile.
If it is actually a cement haze then if you take some diluted vinegar in a out of the way spot and apply it to the haze it should remove it.
Of course rinse the spot with clean water right away and wipe dry.
Etching is corrosion that eats at the marble destroying the shiny polished layer and exposing the more dull marble underneath so you get dull and light discolored spots.
This can kick up a lot of grout dust so make sure the area is well ventilated.
Remove all the grout between the tiles.
In concert with the grout float a large damp sponge can be handy for wiping off any lingering grout haze.
Because it is a polished marble made of calcium carbonate it makes it acid sensitive.
Rinse the sponge often and change the rinse water as it becomes cloudy finally allow.
It is not that hard to repair grout once you un.
As you work be very careful to avoid the edges of the tiles themselves as you do not want to chip or scratch them.
Let it soak for several minutes as directed by the manufacturer instructions.
If you attempt to remove a tile while the grout is in place the grout joints will put pressure on the surrounding tiles which may weaken or crack them.
Apply the grout remover to the grout haze on your marble tile.