The decision had been expected after the terrorist attacks in America, which have damaged prospects for the industry.
Vinci announced on Monday that its offer to buy TBI, for £517m had lapsed. The company owns Luton, Belfast and Cardiff airports.
Vinci, which possesses 15% of TBI stock, required 90% of shareholders to accept its offer before it could make a formal bid. It is understood that Vinci fell slightly short of that figure at the closing date on Monday. This allowed Vinci to withdraw its offer without permission from the stock market's takeover panel.
If it had crossed the 90% threshold it would have been required to go to the panel to argue that it wished to withdraw the offer because of "material adverse change" in TBI's business.