Cook made a defiant 176 in the second innings but could not avert a nine-wicket defeat here.
"If we have to win this series, we need everyone to contribute. There's absolutely no doubt about that. Everyone has to stick his hands up and say that," Cook said in the post-match press conference.
"The lads have not performed as much as they would have liked in this game and they would be very disappointed. That's cricket and we have to bat strong as a team," he added.
Cook said his team will need to improve in a lot of areas.
"When you get beaten by nine wickets you have to look at a lot of areas. There will be a lot to ponder to. There was a small chance of saving the Test, but we could not take that today and that has cost us," he said after his side was beaten with more than one session to spare at the Sardar Patel Stadium.
Though happy to score his third century on the trot as captain, having got two against Bangladesh earlier when he was standing in for the now-retired Andrew Strauss, Cook acknowledged that it was not for a winning cause.
More From This Section
"I am very happy the way I batted, to score any hundred is special for England. To score in that situation made it even more special for me, but the result is really what matters at the end of the day.
"We were not good enough in the five days to win. We showed a lot of character towards the second half of the game. We got bowled out for 191 in the first innings and it was pretty much there we lost the game," said the England opener.