How do you remove marks completely from bathroom countertops?

What do you mean by marks?  Stains or scratches on the surface?  What kind of surface?  More info please.

You tell me that it is "fake marble".  Someone correct me here, but the only fake marble I can think of is resin.  Basically, you have a "plastic" (and I use the term very loosely) countertop.  The vinegar seems to have interacted with the resin and left an etching or stain.

Do you know who installed these countertops?  If so, you need to ask them for advice.  I have ideas but you may not like them.  Do a Google search for "countertop restoration service" or call a countertop sales/installer store and see if they can give a free evaluation.  They'll have to come to the house to check it out.