Jeff, actually one of the Exocet failed to detonate which would've caused even more damage maybe even catastrophic depending on the point of contact. Kudos to the crew for damage control and no doubt their efforts saved the ship however she was probably not combat capable after that with heavy listing to the port side.
The Stark was informed of the aircraft, a Mirage F1, by the local E-2 Sentry when it was 200 miles out. As it got close, the Stark picked it up and tracked it herself. The Stark identified itself twice to the aircraft, and asked the aircraft to identify itself. The Iraqi aircraft never did and then launched, unopposed, from 20 miles out and then later from 15 miles.
For whatever reasons the standard missile system and other defensive systems were offline...which I view as inexcusable. They were never brought up to combat status during the flight of the two missiles.
Anyhow, those two missiles got a free pass.
And you are exactly right. The first Exocet missile struck and did not explode. It hit 10ft above the waterline on the port side near the bridge. Its remaining fuel did ignite and the resulting impact and fire caused significant damage to the ship, causing a fire in the mail room that spread into the Combat Center.
Moments later the second missile struck, also on the port side, and did detonate, blowing a 10 ft x 15 ft hole in the side of the vessel above the waterline there.
The Captain, Captain Glenn R. Brindel, ordered the ship to flood the starboard side to produce a list and keep the large hole well above the water line. It did not list as a result of the hit. The Charles Adams Class Destroyer, USS Waddell, came to the Stark's assistance and escorted the Stark as is made its way, under its own power, to Bahrain. She arrived there the next day and underwent temporary repairs from a US Destroyer tender before returning to the states.
Although her standard missile system was knocked out for the duration, she was able to get her main gun and her CIWS operating. In addition, her Helicopter was also still operational. Despite not having her main defensive battery, she did have combat self defense capability as she retired.
Her crew (which suffered a total of 37 dead and 21 injured out of 176) performed heroically. The ship was designed for warfare and to take hits. It is not known if the 1st missile detonating would have sunk the ship. Since it too hit well above the waterline, there is a a very good chance it would not have, though it certainly would have caused more damage.
So, I view its survival as principally based on the crew's damage control, and the design of the vessel itself foremost. Followed by the fact that the 1st missile did not detonate.
After returning to the States and after the inquiry, Captain Brindel was relieved of command and duty and retired. The Tactical Action Officer, a Lieutenant, resigned from the Navy. The Stark was fully repaired and returned to service.