People like Fauci care most about their reputation/percetion of them by others.
The release of these emails will cause him to be very careful of what he says from here on out. I believe much thought went into the release of this information.
It would not surprise me if Fauci were to resign but I do not believe the President or anyone in his Administration should mention this. So far Trump has done a great job at keeping partisanship out of this crises and it is playing well to Americans across our Country.