King finished fifth in his heat/semi in the early hours of Monday morning in a time of 13.74 seconds.
The top three in each race, plus the two fastest losers, secured a place in Tuesday’s final.
King, whose PB for the event is 13.48 seconds, was just a tenth of a second off reaching the showpiece event.
Afterwards, King said: “Sorry to everyone that stayed up in the UK. I had a bad start and hit hurdles fighting to get back in the race.
“Gutted is an understatement, but I’ll pick myself up and come back stronger. Thanks for all of the support.”
King’s England team-mate Andy Pozzi did reach the final, but his big British rival David Omoregie, running for Wales, could only finish seventh in the second heat in 14.20 seconds.
King’s City of Plymouth team-mate James Arnott will compete on Friday in the T47 100m.