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The hosts collapsed to 223 all out as they slipped to a seventh loss in nine Tests to go 1-0 down in the series.
Former England skippers Michael Vaughan and Alec Stewart have called on the England and Wales Cricket Board to take the captaincy off Cook.
But Cook, who has not made a hundred in 27 innings, said: "To quit now, in the middle of the series, would be wrong."
The Essex opener took charge following Andrew Strauss's retirement in 2012, scoring three centuries in a series win in India and leading England to a 3-0 Ashes success last summer.
Since then, Cook has overseen a 5-0 whitewash by Australia down under and a 1-0 home defeat by Sri Lanka.
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There has been no sign of an upturn in the ongoing series against India, with England going 1-0 down in the second Test at Lord's on Monday.
"I'm not trying to prove anyone wrong," Cook, 29, said. "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. I've been speaking to (head coach) Peter Moores over the last couple of days and, even in tough times, me and him still think we can turn this around.
"A captain is only good as the players you play with. You always think you could do different things in hindsight but I know when we've won games of cricket I've captained well.
"I don't think my captaincy was the reason we lost this game."
The defeat by India also means coach Moores is without a win in four Tests since he was re-appointed in April, but he remained supportive of Cook.
"Alastair has had a tough ride, through the winter and now," said Moores. "The tougher it gets, the more he wants to get stuck in and build this team. He realises that's not an easy job.
"We have to support people and it's great to see he's got the stomach for that fight.
"It's difficult to captain when you're not scoring runs. He's an outstanding player and individual and time will show that. He'll come back stronger."
Listen to Geoffrey Boycott and Jonathan Agnew review the second Test on the Test Match Special podcast.
Alastair Cook said he will not resign as England captain this summer despite the 95-run defeat by India at Lord's.