Direct proof of use
The MIM-23 HAWK was used in the 1990 Gulf War during the opening Iraqi invasion of Kuwait. Redstone Arsenal's official HAWK history states that Kuwaiti air-defense units equipped with U.S. HAWK antiaircraft missiles downed about 22 Iraqi aircraft and one combat helicopter on August 2, 1990.
The same account distinguishes that Kuwaiti combat use from later U.S.-led coalition operations: it says American troops had not fired HAWK in combat and that HAWK was not used by the coalition during Operation Desert Storm.
Sources: HAWK Missile System
Timeline
On August 2, 1990, Iraq invaded Kuwait. Kuwaiti HAWK batteries formed part of the short-lived Kuwaiti air-defense response during the invasion, and the Redstone account credits them with multiple Iraqi aircraft shootdowns.
After Iraqi forces overran Kuwait, captured HAWK and TOW missile stocks in Kuwait were described by Redstone as a possible threat to the U.S.-led coalition. The broader campaign then shifted into sanctions, coalition buildup, and Operation Desert Storm, which expelled Iraqi forces from Kuwait by February 28, 1991.
Sources: HAWK Missile System, State Department Gulf War
Narrative
HAWK's Gulf War role was concentrated at the start of the conflict rather than in the main January-February 1991 coalition air campaign. Kuwaiti units used the system as ground-based air defense against Iraqi aircraft during the invasion, while coalition forces later treated captured Kuwaiti HAWK stocks as materiel that could potentially threaten friendly aircraft.
The documented user was Kuwait, represented in this catalog under the Gulf War coalition side. The available official source identifies the weapon, operator, date, target category, and captured-stock concern, but does not break down individual launch sites, battery identities, or aircraft types for each claimed engagement.
Sources: HAWK Missile System, State Department Gulf War