Gruden out Agreed. I have a hard time mustering up a defense of a celebrity doing something that would get the average worker fired/disciplined/shunned in their workplace, and I think using slurs to talk about members of your organization and sharing topless photos of coworkers in documented conversations to a supervisor would qualify.
And ignoring the morality of the above (not that one should), it's a serious lapse of judgement when you're paid millions in the spotlight to do anything but maintain a squeaky clean record when you're talking to people in the organization. Based on the kind of person Gruden seems to be in private and his judgement calls, I personally wouldn't want him to work for me if I had the choice, nor would I think him qualified to be a leader on the field and in the community based on that either.
Edit: Changed to "a supervisor" to reflect that it wasn't his team's owner. Point still stands that technically all of the NFL is the same organization, especially since some of the slurs were directed at the commissioner himself