Ireland took an early lead on Sunday through Robbie Brady's penalty but faded in the second half, eventually losing 2-1 to an Antoine Griezmann double.
With France having six days to recover from their previous outing compared to Ireland's three, O'Neill criticised the organisation of the competition.
Asked whether the difference in rest periods between the two sides impacted the outcome, O'Neill told ITV: "It will sound from here now as if that's an excuse, but it's not meant to be. It is incredible. Absolutely incredible.
"We knew at the start of the tournament that, if we took our place in the last 16, the chances are it was going to be a big differential between us and some of the sides we might play in the competition.
"It looked as if it was geared for them [France] to win their group, which they did. Well done to them, but three days [more rest] is too much.
"It's genuinely far too much and it took its toll on us in the second half."
However, O'Neill was proud of the effort his side produced in the tournament.
"The team put absolutely everything into the games that we played," he added. "The only disappointment we had was the second half against Belgium.
"Outside that, it's been a phenomenal effort.
"Physically, it was very, very demanding today. We got the goal and we were pretty comfortable in the first half.
"Perhaps if we'd gone another six, seven, eight minutes without conceding, I think there would have been a lot of pressure on France and we could have used that to our advantage."