Nope, that's unacceptable. There are too many stories out there about freak accidents that occur with no warning that leave a child injured or worse. A quick google search about falling bookcases and chests of drawers falling on kids is enough to scare any sensible person.
Scary statistic in subheading and a good list to refer to:
http://www.goodhousekeeping.com/home/renovation/a33636/home-hazards-for-children/
Aside from the whole childproofing thing, the nanny's job is to create a safe environment for the child. Unless she has a monitor as well and is actively viewing the screen during this time, then your child is a bit too young for the nanny to leave for so long. Independent play is extremely important, but not at the risk of your child's safety. My first thought was that the nanny was putting other things in front of your child as priorities, when it sounds like you have done everything in your power to assure that she has no pressing concerns other than caring for your child while at work (kudos for this).
I would casually talk to nanny about how nanny thinks the child's independent play time is going and then see how she responds to that. Tell her your preferred method of allowing your child this time, mention that you don't feel comfortable leaving him alone for more that X amount of minutes, and check and see if she follows suit in the coming days. Us nannies learn from the parents and if yours is not amenable to your methods then it sounds like it's time to find a new nanny.