No modern navy has entered a significant nautical engagement in over 40 years with the exception of the Falklands.
The Argentine air force managed to do quite a bit of damage with air-2-surface missles. There have been many naval studies conducted regarding closue of the Hormuz Straits, as well as the Mallaca Straits, and in every simulation blue forces take considerable losses.
I'm not suggesting the USN would not suceed. However, people need to understand the reality of the situation. These are not blue water operations. The Hormuz is extremely narrow and there are 100s of miles of coast line for silkworm batteries as well as portable fired surface to surface platforms.
In addition, our air force would need at least 1-3 weeks to establish clear approach lanes for our air assets to handle surface interdicition from the air. You do not want to bring cruisers or even frigates into such a confined area. That would be insane. I don't know if you remember the Stark incident, but this debacle shows the complexity of operating in such a tight environment and the quick decision making required for operations like this.
If you remember the Falklands campaign by the Royal Navy, you might remember that a few Argentian Mystres wewre able to breach the air cordone and do quite a bit of damage to her Majesties fleet using anti-shipping missles. When you have a sub-sonic, extremely small cylinder traveling in terms of meeters off the water, they are very hard to detect. And the warheads in these devices are extremely powerful.
One of a captains worst night mare is a small attack craft close to the shore and unvisible due to land and small craft clutter to be firing these types of weapon systems.
I can assure that any sea-2-sea engagements inside the straight would be spearheaded by subsurface systems and air. An Arleigh Burke will be hard pressed to utilize its full capabilities in such a tight cordone of operations.
An all out attempt to close the Hormuz straigts by Iran should not be brushed off ligthly. It will be a costly and potentially protracted effort that will require our political leaders to pull out a lot of stops and dispense with our reluctance to use a wide array of high tech classified technologies.
Did the brits have CIWS (Close In Weapon System) during the Falklands?
As for weeks for air force, um, the Navy brings its own. As for very restricted waters, you can fire from outside the straits, easily. US surface combatants have Tomahawks. And their primary air defense umbrella extends out to 100 miles. The main worries are (1) mines, which the Iranians would lay by sub and by small boats operating at night, and (2) shore batteries of SSMs. But the second of those can be shot out of the sky by modern air defense, which all our DDGs and CGs have. Of course you'd also want to go after the launch sites, with F-18s and Tomahawks.
On mines, we are much better prepared than in the 80s because of the lessons then. The first line is all the helos the surface ships carry can deploy sweeping gear. There are also a number of special mine warfare ships prepositioned in the area.
Might a US surface combatant take a hit occasionally? Sure, it is possible. All that have in recent times have survived, and we aren't going to run out. Won't seriously hamper operations at all. The Japanese hit 500 US ships off Okinawa and never stopped a single operation. The USN does not stop for donuts.
A more realistic worry is the shore battery SSMs, mines, and perhaps also their few subs directly, hitting civilian shipping in the gulf, rather than warships. So you have to take out those toys and you have to escort shipping in the meantime with air defense capable ships. All do-able, with some period of disruption of traffic, obviously. That is the only cost the Iranians can seriously impose.