I’ve seen both the original and the Clooney remake. Let’s see if I can recall what did happen.
U.S. orders their fighters to intercept and if necessary shoot down the bombers, but the fighters ran out of fuel before they could reach the bombers.
Both the President and the group commander’s wife personally appeal the crews to break off the attack, but the commander is trained to ignore any communications past X point.
U.S. tells the Soviets to ignore one unarmed bomber and concentrate their defenses on the bombers carrying the nukes. The Soviets think it’s a trick, continue trying to get all the bombers on the attack run and wind up shooting down the wrong one.
U.S. tells the Soviets how to jam the bombers’ electronic defenses, but the bomber crews manage to overcome the interference.
Obviously these points were covered briefly in the movies. I didn’t read the book, so I don’t know whether there was a longer explanation of why each of these tactics failed, or a longer discussion of whether the Soviets were deploying their own bomber forces, missile-launching subs, etc. In any case, the plot was set up to leave the President with three options:
Do nothing and let the Soviets take whatever revenge they saw fit after Moscow got nuked
Launch an all-out war on the assumption the Soviets would do the same
Make a huge sacrifice to atone and hope the Soviets accept it and go no further.
Doing nothing was not an option, as the Soviets would have retaliated. An all-out follow-up was dismissed, so the question really comes down to what sort of symbolic atonement would be sufficient.
From a literary/dramatic standpoint there are only two options – we nuke one of our own cities (it really doesn’t matter whether it’s New York, Washington or wherever) or the President goes on TV, takes responsibility and commits suicide.
If the President commits suicide, then there’s no one left to negotiate a treaty with the Soviet premier (who also survived) to prevent a future mistake and the ending is even more bleak. Therefore, the only option was for the President to sacrifice a city to save the world.