Kane's decisive penalty miss proved crucial as France prevailed 2-1 at the Al Bayt Stadium on Saturday.
The England captain had earlier in the second half sent one spot kick past Tottenham Hotspur teammate Hugo Lloris in the France goal to cancel out Aurelien Tchouameni's first half opener.
But after Olivier Giroud restored the defending champions' lead in the 78th minute England were awarded a penalty to restore parity after Theo Hernandez bundled Mason Mount over in the box.
Kane looked Lloris in the eye and sent his effort high over the bar as England exited the World Cup at the quarterfinal stage for the seventh time, more than any other side in the history of the competition.
"It's tough to take for me personally and for the team," Kane told BBC Sport after the game.
"I couldn't be prouder of the boys. We had better spells, better chances, but football comes down to small details.
"As the captain and the one who missed the penalty, I take responsibility for that. I can't fault my preparation or the details before the game.
"I felt confident taking it, I just didn't execute it the way I wanted to. It will be something I have to live with and take on the chin.
"The team is in a really good place and there will be highs in the future. Standing here now, we are gutted it has come to an end as we had full belief we could go all the way. We can be proud of what we've done."
