So recently, I've lost all vision in my right eye due bad luck in the genetic lottery and was interested in getting my first compound bow.  With my handguns, I've been able to stick to my natural RH dominance with some minor adjustments to my stance, but with my long rifles, I've had to switch over to shooting with my "weak hand" and relearning how to sight and shoot with ease and comfort. That being said, do I need to look at using a left handed bow? Or can I use a right handed bow and alter my stance/grip/sight to use properly?

With a compound with sight, I'd go with a left handed bow.  Your head position, anchor point with a right would be very inconsistent.  Might consider a longbow, or recurve, and shoot without sights.

I would go with a left hand bow if you plan on using sights with pins.   They do make laser sights for bows, and red dot sights.   If you could use those with your left eye, you could shoot a right handed bow, but it may be better to relearn with a left handed bow for greater accuracy so you can use a peep sight on the string.

Go with a left handed bow! My wife does not have vision lost, but she is right handed with left eye dominate. When shopping for a bow all the pro-shops suggested going with a left-handed bow for her.

If this is indeed your first bow and you're not really used to shooting one way or the other, I'd go with a lefty.  Moving your head around in order to shoot right handed would probably make it a little harder to acquire solid and repeatable anchor points anyways.

Well if you have lost sight in your right eye I would recommend using a left handed bow. My question is how accurate you will be without the depth perception of both eyes.

Spot on in above post. If your not dead set on a vertical bow, go with a crossbow.