The first thing you should know is that you are not going to lose your driver's license. Pretty much all of the states allow one-eyed drivers, so long as their eye is correctable to close to 20/20 vision. You probably want to check with your state's motor vehicle department for any special restrictions or rules, but by and large your state probably won't care that you only have one eye (unless you try to drive a big rig or a school bus).
The next thing you should know is that outside of about 20 feet, everybody sees the world as if they only had one eye. This means that your driving will be pretty much unchanged for everything that you see which is more than 20 feet away from you. But you do have to watch out for two things:
1) You will not have the same peripheral vision as you did before, because of your nose. This means that you will have somewhat of a "blind spot" on the side of your lost eye. It will take a couple of days to get used to this blind spot, and you will need to learn to turn your head more frequently from side-to-side instead of merely relying on sideward glances from your remaining eye.
2) Within 20 feet you will not have the same level of depth perception as you did before. This means that you must be especially careful in parking, and you should avoid "tight spaces".
I suggest that you mount some "fish-eye" mirrors on your side rear-view mirrors, to help eliminate these blind spots. You might also try one of the large wide-angle rear-view mirrors.
I also suggest that in the future you select vehicles with excellent fields-of-view and few blinds spots. After my surgery, I immediately sold my sportscar and instead purchased a Jeep Cherokee/Ford Explorer-sized SUV which offers excellent visibility and is very easy to corner and park. I would also avoid, at least at first, large automobiles because you might find them difficult to park.
Obviously, you should take greater pains to protect your good eye while driving since if something gets into your good eye you won't have many choices except to slam on the breaks until you can get it cleared.
Except for these two concerns, your driving should be about the same. I suggest that you get out and drive as soon as possible after the loss of your eye, to psychologically reassure yourself that you will retain the same mobility as you had before your loss. For me, it was a big mental step towards recovery to get out and drive myself to work and back, and figure out that I can still go where I want to go when I want to go there.
But don't be careless -- for your first several trips please take someone with you to help you with the blind spots and measuring close distances while parking (this is more for your confidence -- you'll find in the first 20 feet that you don't need any help and will be just fine).
Handicapped Plates/Sticker -- The California Department of Motor Vehicles issued to me a handicapped sticker, which is sometimes useful for parking, Where it helps the most is NOT because of any depth perception concerns (which is probably why they issued it), but rather having a slightly reduced field of vision, it helps on the safety side when backing out.
As one-eyed commercial pilot Frank B. Brady recounted in his must-read book A
Singular View: The Art of Seeing With One Eye, in 1992 the F.A.A. reported
that it had licensed 218 first class, 739 second class, and 2623 third
class medical certificates to one-eyed pilots.