They're obvious enough that I don't think describing them is a problem.
1. Like ballistic missiles, there is a vulnerable "boost" phase, the subsonic flight until within ten miles of the target. The more distant the launch vessel, the longer this phase is. The closer the vessel is to the target, the more likely it is already being tracked.
2. A rocket-propelled system cannot be made stealthy, and unlike a ballistic missile which "coasts" for most of its flight, a surface-hugging supersonic rocket must be under propulsion for its entire flight. Tracking this missile in its terminal phase is not going to be difficult. Hitting it might be, since it has evasive capability.
Whether or not the Navy will quickly come up with exploits of these flaws is unknown. I can think of a few, including weapons based on aircraft overlooking the area, and nothing exotic like lasers would be required. The key is to shoot it down before the rocket phase.
A description of a carrier's outer defenses from Wikipedia:
A standard formation provides a number of layers of defense, designed to give maximum protection to the fleet's high value units (HVUs) or main body. Furthest out are the picket ships, Combat Air Patrol (CAP) craft and early warning aircraft (AEW). These units operate at 200 nautical miles (370 km) or more out from the main body. The units of the outer screen operate between 12 and 25 nautical miles (22 and 46 km) from the main body. The inner screen is within 10 nautical miles (19 km) of the HVUs.
I understand the special circumstances of Gulf operations. One obvious tactic that would make the missile more effective is a shore launch while carriers are forced close to the launch point, i.e. in or near the Straits of Hormuz. If the fleet is around ten miles from the shore, the missile would be hard to stop using conventional weapons, and a hidden shore launch battery would be very hard to detect. Obviously, missiles would be launched from many separate locations, not all from the same place.
In the open sea, there would be more options.