Actually, according to the WP article sources they did.

There were Americans on nearby warships who knew (and arguably, should have provided forceful backup; but the other warships had far inferior radars and thought that Vincennes knew something they didn't). The watchstanders on Vincennes did not know. They should have known, but there was no "bright line" that makes a given radar target look automatically civilian or automatically military on the display. Watchstanders and the CO have to make that determination based on a confluence of factors, including IFF transponder frequency, flight plan, whether the craft is ascending/descending, etc. The CO came to the conclusion early that the indications he had were of an attacking aircraft. Part of that was based on a determination that the aircraft was descending in a hostile attack pattern. It wasn't, and the AEGIS radar showed as much. But when the life-or-death decision point for your ship rests on whether you see a '-' or a '+' next to a altitude change rate it's not astonishing when a watchstander makes the wrong call.

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.