Yes the IOWA class would have been damaged. They would have likely remained mission capable. Analysis done years ago suggested multiple hits would be required to defeat a large ship with defensive armament. Iowa class were very vulnerable to breaking the keel type detonations and deck strikes.
The real question is why did not CIWS not defeat this rather ordinary threat?
The IOWAs could have been damaged IF they'd been hit. Not only can they take damage much better but they should also be able to defend themselves better. How many CIWS would fit on an Iowa? If the system is really flawed (rapid and accurate analysis is essential here!) than more wouldn't help, but if the system is just not quite as good as expected then multiple guns would provide much better odds. And their longer stand off range gives you a much longer time to react to an incoming threat. If you were willing to trade throw weight for range with the advanced munitions developed in the late 60s you could probably out range this missile with a similar payload multiplied by nine shells every couple minutes. Add modern GPS packages and fire control drones and the Iranian rocket crews are going to be volunteering for safer duty as suicide bombers.