I don't believe we ever do, but this doesn't need to be taken literally.
I don't remember the context in which this was said, but it's another way of saying something like "I've cared as much as I can, there isn't any more I can give" or "I've worked through the pain already and now it is time to move on" or "it isn't worth the pain of caring deeply." He doesn't need to literally cry to feel deep sadness.
It's like the Loreal Kids shampoo deal. "No more tears" was the slogan, but it still hurt your eyes. They claimed "tears" was like tearing your hair, not tears in your eyes, even though in the commercials they pronounced it "teer" and not "tare."
Well to be fair, half of the games in the series have a different Snake. There is only one more game in the series that takes place after that with Snake as the main character (MGS4).