England slumped from the seemingly comfortable position of 173 for four shortly after lunch to 223 all out before tea, with poor shot selection rather than excellent India bowling extending their sequence to 10 Tests without victory.
Ishant Sharma accounted for Moeen Ali, Matt Prior, Ben Stokes, Joe Root and Stuart Broad on the final day to record figures of seven for 74 and, while Cook admitted his side lack confidence to take chances to win Tests at present, he insisted he is not about to step down.
He said: "It gets harder and harder the more we don't win and of course that heaps on you at the end of the day. Until that bloke taps me on the shoulder and says 'we don't want you to captain', I'm desperate to try to turn this around for England. If I'm not good enough at the end of the summer then so be it, but I'm trying my heart out to do this."
Cook went on to give his backing to under-fire wicketkeeper Matt Prior, who has come in for criticism for the nature of his dismissal when England needed him to defend his wicket.
"Matt's got a serious amount of talent; he's been a fantastic player for England," said the captain. "He's desperate to carry on playing and, at the moment, I think he's the best wicketkeeper-batsman in the country. Of course he has to always prove [it with his] performances - that's the way the world goes and every single guy has to earn the right to play for England."
Among those heavily critical of England in the wake of their defeat was the former captain Michael Vaughan, who called for change after labelling the performance at Lord's "pathetic". Several batsmen were undone by the short ball, with Prior, Stokes and Root giving away their wicket by hooking Sharma to catchers deep on the legside.
Following defeats in seven of their last 10 Test matches, their latest loss led to Vaughan reiterating his call for Cook to be replaced as captain. He said: "It's been absolutely pathetic from England. We've seen some collapses in the last year - against Australia, Sri Lanka and now India - but this is the worst of the lot. Something has got to change.
"India were the better team over the five days; they were better led, and batted and bowled better. What was said in the England dressing room at lunch? To come out and play all those shots, it's ridiculous.
"It's as though Matt Prior came out thinking he was in the last-chance saloon and was going to hook every single ball. You can't do that in a Test match. Ben Stokes played a hook shot on a pair. It was pure panic - but why are they panicking?"
Andrew Strauss, Cook's predecessor as England captain, said the hosts' failure to capitalise on winning the toss on a seam-friendly wicket came back to haunt them. "James Anderson's run-out was an apt finish for a shambolic end to a shambolic Test match as far as England are concerned," he said. "They would have come here with very high hopes of winning especially when they won the toss. They fluffed their lines all the way through the game. There are some good players who've given a lot to England over the years but at the moment they're just not at the races."
Cook's own form is an ongoing concern, although Nasser Hussain believes there are too many fundamental things going wrong for England that will not be resolved by Cook regaining his form.
Hussain said: ""If Michael Clarke was suddenly not captain of Australia they would miss him. If Alastair Cook was not captain of England at the Ageas Bowl would we look up and say 'we're missing the tactical genius of Alastair Cook?' We wouldn't.
"There's been a lot of stuff written this week that he just needs a score. That will not change all of England's woes. If he gets an 80 at the Ageas Bowl, will they stop collapsing? No. Will they bowl better to the tail? No. Will they bowl better with the new bowl? No. Will they play the short ball better? No. It's a sign of madness if you keep doing the same thing time after time and not changing it."
Man of the match Sharma, meanwhile, was quick to pay tribute to his team-mates. "The way the batsman played in the first innings was tremendous," he said. "Ajinkya [Rahane] played really well to score that hundred. That gave us the momentum to bowl in the second innings to get them out cheaply."
He added: "I don't think all the wickets are for me; they are for the captain because he's the one that told me, 'you are tall enough so you have to bowl a bouncer, you have to use your height', and that's what I did and it worked for me."