It would quickly involve US forces as the first US forces were heading Saudi Arabia by 8 August 1990 primarily to protect airbases. The ready brigade of 82nd Airborne would have been there within 48 hours.
www.nytimes.com
Obviously these forces were not enough to stand up to an advancing Iraqi corps but they were a gut check force to see if the Iraqis were going to escalate the war.
I guess the invasion depends on the hearts of heart thing.
If Iraq sweeps in and the Saudis fold before the 82 come and the royal family is already, maybe some sell out or dead then things get awkward.
While I doubt Iraq would shot first I also doubt the airborne would feel bold about fighting something already lost, then again it depends on their commander.
Granted I'm not saying they would do nothing, the UAE, Bahrain, ect would very much like a wiretrap of US troops if Saudi Arabia does fall quickly.
So they could be redeployed.