The missing eye is just a folktale perpetuated by vidya games. They actually recovered his body and his right eye was there, albeit decomposed to hell.

Oh thats a bummer, but the video games just made it popular I guess, the tale was there before...but I guess it makes sense, in times of war, creating rumors were used to affect morale of allies and scare enemies...the whole "One-eyed dragon" deal...oh well, he is still cool and reckless (still not as much as Oda that crazy man)

Date Masamune actually did only have one eye. His right/left eye (im not sure which side) was lost to smallpox as a child so he could only see out of the other eye. Later on the eye was removed entirely, but no one knows how exactly it happened. As rogue said, they recovered his body and the right eye was almost completely missing; it only retained a few organs.

There's a story that he had one of his retainers gouge it out so he could eat it because he thought it would be a shame to throw it away since it was a gift from his parents. I'm not really sure how true that story is, because it's very similar of the romanticized tale of how Xiahou Dun lost his eye (from Romance of the Three Kingdoms which was popular in Japan at the time, he removed the eye mid-battle and ate it to frighten his enemies), but there you go.