As far as I know, the "Jack story" ran. Olivia even made a comment last night as to how she'd already given the reporter one story that day.
Would Ethan (chief of staff, not Sec. of State) un-resign as a result of the story being false? No. That would be up to the President, and generally you want to have someone in your administration volunteer to be thrown under the bus when you have a day of disasters, as this day would have been for the Taylor White House.
By resigning, Ethan has given outsiders someone to blame for the problems of the day. Of course, Ethan would normally stay around the administration in some fashion since he resigned as a result of something that didn't really happen; he'd just be placed in a different role.
And Allison Taylor wanted to keep her daughter around. She fails to realize that she failed as a parent even more than she has failed as president.