What would be the designation of female president's husband in the USA?
There isn't an official title for the position (just like "First Lady" isn't official - it's merely the convention used to refer to a president's wife), the most likely designation would be "First Gentleman".
Given that the wife of a male US president was termed "First Lady", there would need to be a term for the husband of a female US president that held the same timbre as "lady" does for a woman. I think the prevailing term used by the media (and it certainly wasn't and isn't official) was "First Gentleman" before the election results made the question moot for at least four years.
There were other references to "First Man", "First Husband", and "First Dude" and others as well, but none of them would have held the gravitas that this particular title would need.