Every GIA report carries a number. Enter it on GIA's own Report Check service — free, public, and hosted on the institute's website — and it returns the grades GIA issued against that number. Compare every field: carat weight, shape, cut, colour, clarity, measurements and any comments.
Most GIA-graded diamonds above a certain size also carry a laser inscription on the girdle: the report number, in microscopic characters, readable under magnification. A jeweller can show it to you in seconds. Matching the inscription to the report is what closes the gap that verification alone leaves open.
If the number returns nothing, the report is not a GIA report. If it returns different grades from the ones printed, the printed report has been altered. Either finding ends the transaction.
GIA also grades laboratory-grown diamonds and states origin explicitly on the report. A report that does not address origin at all on a modern stone is worth a second look, because origin disclosure is now standard.