Multiple sources of (dis)advantage don't "stack" (PHB p. 173, Player's Basic Rules p. 57, emphasis mine):
If multiple situations affect a roll and each one grants advantage or imposes disadvantage on it, you don't roll more than one additional d20. If two favorable situations grant advantage, for example, you still roll only one
(Dis)Advantage can't make you roll more than 1 additional d20. Having 2×disadvantage is exactly the same as having 1×disadvantage. This is true even for advantage canceling out disadvantage: 2×disadvantage + 1×advantage cancels out completely, rather than leave you with 1×disadvantage. The concept of (dis)advantage is designed to be as simple as possible, and any stacking rules would possibly negate this advantage of the system.
So, yes, that fighter could close his eyes and face no mechanical repercussions from it and still gain the benefit of avoiding the Basilisk's gaze. However, the character normally isn't aware of the rules and therefore doesn't know that. This sort of meta-gaming is often (but not necessarily) discouraged by groups. Unfortunately, it can be hard to avoid.
The key question to pose to the player here would be "If your character were not exhausted, would he avert his eyes?". "Yes, of course, that gaze is nasty!" can be a perfectly valid answer, in which case the player benefits from the simplicity of the rules. Just make sure that the player's justification for his character's closing the eyes is not just "I already have disadvantage, so I don't lose anything!"