The shootdown of Iran Air 655 occurred during Operation Earnest Will, which involved the convoying of Kuwaiti (not Iraqi) oil tankers through the Persian Gulf to protect them from Iranian interdiction contrary to the UN Convention of the Law of the Sea regarding international waters and navigation rights.
Throughout this operation, Iran conducted mining efforts within the Persian Gulf. On the very first convoy an oil tanker was hit by an Iranian mine (luckily causing no injuries or loss of life). Later in the convoy operation U.S. special operating forces managed to capture an Iranian minelayer in the act of laying mines.
Later during this same operation, USS Samuel B. Roberts (FFG-58) was struck by an Iranian mine (proven based on serial numbers matching mines noted from the aforementioned spec-ops mission).
The mine caused damage which is normally fatal to warships of that size (the keel cracked and both engines were knocked from their mounts). She had to limp out of the minefield on auxiliary thrusters while the crew fought to contain damage and save the ship. 10 sailors were injured badly enough to require evacuation. 4 of those were badly burned and had to be sent all the way back to the U.S. to recuperate.
As a result of that, the U.S. Navy launched a short surface operation against Iranian platforms (Operation Praying Mantis) on April 18, 1988.
Meanwhile, about a year before Iran Air 655, on May 17, 1987, an Iraqi (U.S. ally??) Air Force jet had attacked the frigate USS Stark (FFG-31) with two Exocet missile in the Persian Gulf. Though the damage to the ship itself was not as severe, 37 sailors died and 21 more were injured.
In the context of all of that, on July 3, 1988, USS Vincennes shot down Iran Air 655. To be clear the ship captain was clearly in the wrong. Even with the 'fog of war' and the problems of having to use instruments to fight instead of one's own eyes he had enough indications to know he wasn't under attack.
But the shootdown itself didn't come out of thin air, and the U.S. was brought into a tense situation in the lead-up to the events of that day (though admittedly, they didn't do much to de-escalate either once it started happening).
People died all the time on the interface between the First and Second Worlds, yet it was still termed a 'cold war' for the decades it existed.
And a hot war is one where bullets, missiles, shells are flying around. Just because there's no declaration of war doesn't mean it's not a war. Let's look at a little history here:
Korean War - Hmm, no declaration of war Vietnam War - Same Grenada - Same Panama - Same Desert Shield/Storm - hmm, no declaration here either OIF - Same OEF - Same
So the pattern is pretty clear, that declarations of war aren't worth the paper they're printed on. And try telling the combatants on both sides in the above conflicts that they weren't in a hot war.
And a hot war is one where bullets, missiles, shells are flying around So the Cold War was a hot war then? You can't fix a poor definition with a poorer definition.
In this discussion it hardly matters who was on the right side of the overall conflict, however, in this specific engagement: the Iranians shooting at the Americans was reasonable and should not have surprised anybody; the Americans shooting back was reasonable and should not have surprised anybody. The only real problem was the American shooting at the airliner.
The Vincennes watchstanders tried hailing the aircraft two times, but didn't get a response. But even that didn't really help resolve the issue; was it an hostile aircraft maintaining radio silence or a confused civilian airliner? They used a frequency a civilian airliner could have picked up, but it turns out there were several other frequencies the airliner could have been monitoring as well, and besides... how was the Iran Air 655 pilot to know Vincennes was ever referring to them?
Vincennes was having to make a hail to the effect of "unidentified aircraft south of Bandar abbas with speed 350 knots, heading 210, you are on a hostile course and this warship will take defensive measures". What civilian airliner pilot would think that applied to them (though Vincennes quoted speed-over-ground and not the airspeed the pilot would have been looking at).
So there were tons of 'Final Destination'-esque problems that occurred here, but foremost of all was that the watchteam (CO and CIC watchstanders both) determined early based on a single misread indication that they were being attacked, and the evidence they had available to them after that could easily be construed to support that determination.
The entire operation (and several other operations close in time in the Persian Gulf) was the US administration choosing to directly intervene in support of Iraq in the war with Iran; the limited scope of the intervention was a matter of sell it, above all to US domestic audiences, as something other than the direct support for Iraq that it was.
And if the U.S. wanted to re-enact a modern form of the old "Cash and Carry" plan they imposed on the U.K. before Pearl Harbor, then they were within their rights to do so as well as long as they obeyed UNCLOS.
The additional U.S. direct intervention on the side of Iraq (e.g. weapons, training) has been pretty conclusively demonstrated to have been stupid and unhelpful with regard to geopolitical stability in the Middle East (several times over, by now).
But in any event, if Iran had really wanted to blockade Kuwait then they should have declared war and blockaded Kuwait (it doesn't exactly have a wide coastline), instead of mining international waters. At least then they would have the moral high ground should the U.S. still have decided to intervene.
"The lack of international sympathy disturbed the Iranian leadership, and they came to the conclusion that the United States was on the verge of waging a full-scale war against them, and that Iraq was on the verge of unleashing its entire chemical arsenal upon their major cities."
The aftermath of the incident has no bearing on whether the Iranian reaction to the intrusion was rational or justified. The order of events does not support such a line of reasoning. What you are basically saying is that nobody should ever shoot at Americans no matter where they are or what they are doing, no matter the current military or political circumstance. Ridiculous.