The unpopular Domenech was heavily criticised for France's poor showing in their 2-1 aggregate play-off victory over Ireland, especially after their 1-1 draw at home in the return leg.
William Gallas scored the equaliser from an Henry cross in extra time after the striker had handled the ball.
"I have never resigned and I never will. The more I am under pressure, the more I am motivated," Domenech said in an interview published by French weekly L'Express on its website on Tuesday.
Asked if now would not be a good time to make a gracious exit, Domenech said: "People would think: 'What a chicken, he is fleeing'.
"Leaving now wouldn't make any sense. It would mean I've done my job and I don't give a damn about what happens next, it would prove my critics right. People must stop dreaming: I am not the kind of guy who gives up," he added.
Henry has been accused in France of setting a bad example to young people.
"Why do you want soccer to be a moral matter?" Domenech asked. "Of course, you must abide by the sport's rules but (football) is also a reflection of society. Highbrows say they are outraged but, as far as I know, Thierry did not commit a crime."
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