“I am absolutely devastated for those GB men,” Kate Richardson-Walsh, gold medallist in Rio in 2016, said on BBC TV.
“This team is absolutely capable of being Olympic medallists, I have no doubt about that, but we are going to have to wait.
“Four years seems a long way away now.”
Great Britain have not won a men’s Olympic hockey medal since their gold at the Seoul Games.
They came into Paris as the world’s second-ranked side and were tipped as medal contenders – which will make this defeat all the more painful.
The side spoke about their never-say-die attitude in the pool stages, where they came from behind in three matches in a row to twice draw and then win.
GB were the dominant side against India, particularly after Amit Rohidas was sent off.
Red cards are exceptionally rare in hockey but the defender was shown one after just 17 minutes when he caught William Calnan in the face with his raised stick.
India kept their composure and took the lead through a powerful swept shot from Harmanpreet Singh that flew past goalkeeper Ollie Payne.
Morton was in the right place to pounce and shovel in his third goal of the Games to draw level.
GB pushed hard for a winner but India goalkeeper PR Sreejesh produced a stunning block to deny Calman in the closing stages and Sam Ward was unable to poke home from close range.
That led to a shootout and, with much of the crowd supporting India, Sreejesh saved attempts from Williamson and Roper.