"Position has become untenable" is a bit of an over-used phrase for any public figure that's subject to intense scrutiny, but surely has to apply here. I mean they've literally got the whole squad (and beyond) saying they won't play whilst he's in place, so regardless of whether the individual concerned is deemed to have been guilty of wrong-doing or not, that's an impossible scenario to continue with.
By the sounds of it this is just uncovering a toxic culture behind the scenes that was bubbling away (hence the refusal of many to play for the coach), what's interesting is the claims that those in power tried to influence Hermoso to hold off, to issue statements backing the president etc. Reminds me a bit of the situation with that Chinese tennis player who made some sort of sexual allegation against a politician and then disappeared and was forced to put out a video saying "I'm fine innit" or something (memory a bit hazy).
The issue the Spanish FA has is this transcends international boundaries and has a lot of scrutiny so it's hard to brush under the carpet in the way they might be able to with a situation at a domestic team. They could have a scenario where other countries shun the Spanish FA until they sort it out, I mean technically they could bar Spain from playing in the Nations league I guess, assuming they could even find the players willing to play?
edit: I'd like to add this must be pretty bloody annoying for the players (and Hermoso especially), even leaving aside the wrongness of the incident, you've won the World Cup and now this has stolen all the limelight, you can't properly enjoy the moment. Heck there will probably be events cancelled, the winners will get invited on chat shows and then this nonsense dominates proceedings etc (contrast with England last summer where some of the girls probably took the opportunity to leverage their fame and earn a few quid etc, good luck to them).