What Solutions are Available for Marble Floor Etching / Corrosion
That's too deep to grind flat.  It's somewhat of a "natural" look, so you could consider leaving it as-is unless the holes are big enough to be a hazard.  In that case, I would look at just filling the holes. 

You can use clear resin, which will sort of keep the natural look.  A variation is to fill the holes with a mix of resin and marble dust.  If you can find a small piece of marble of a generally similar color (doesn't need to be an exact match), you can make your own dust with a grinder.  A marble supplier may also be a ready source of dust from other jobs.  That will look a little more like natural patterns in the stone (like the light colored blotches that are part of the existing stone).  

When it hardens, you grind the filler level and polish it.  Experiment first to learn how the material behaves, and adjust your mixture to achieve the appearance you want.  Buy a few marble scraps to play with.  You can use a Dremel tool to grind out some holes to mimic the floor and practice on that.  Work with small batches so you have plenty of time to get each repair right.

You can also buy kits for repair of marble and other stone.  They tend to be epoxy with colorants that you mix to match the color, or an acrylic gel that you color.  Here are a few links (there are lots of options that a Google search will show): 


https://www.homedepot.com/p/Custom-Building-Products-Travertine-Repair-Kit-TRAVRK/203919972
https://marblekareproducts.com/products/mk-hole-repair-kit-white?variant=553746433
https://countertopguides.com/shop/countertop-stone-repair/granite-marble-chip-repair-kit
http://acrylicrepair.com/high-quality-do-it-yourself-marble-repair


Disclaimer: I've never used any of these products, they're just examples as illustrations.  I mix my own.

If it's just a few tiles, you might be able to replace them from a marble supplier.  You won't find an exact match, but you might find something close enough (or what you think is a better match than a few tiles with patterns that don't match the others).

One other thought: If you use the resin and marble dust approach, that material will be almost a tough as stone and you may need an angle grinder and a collection of successively finer grinding heads or pads to smooth and polish the surface in a reasonable time. If you use one of the kits that are just colored resin, that material is much softer. It will scratch more easily, but you will be able to finish the surface quickly with a collection of successively finer sandpapers. 

With the resin-only approach, you can put down a shine layer on the whole floor.  There are products like Pledge Floor Care that are a clear acrylic solution that dries to a film. That will get the abuse and you just periodically clean it off and apply a new layer rather than polishing away the repair over time, which would eventually leave dents.