The Three Lions suffered a resounding defeat on Monday to go 1-0 down in the five-Test series with three matches left to play.
Needing 214 runs to win with six wickets in hand at the start of the final day, a batting collapse saw the home side lose to India at Lord's for the first time in 28 years.
"I'm not trying to prove anyone wrong," Cook told Sky Sports News. "When I was made England captain, I told my wife I would give it my all. If it's not to be, so be it but I believe the team needs me to lead them through this tough time.
"If it gets to the stage where my position becomes untenable, I will be honest to myself. To quit now, in the middle of the series, would be wrong. I'm desperate to try and turn this around."
England and India start their third Test at The Ageas Bowl on July 27.