Josh Sargent struck twice to lift Norwich out of the relegation zone as Watford finished the game with 10 men, piling more pressure on Ranieri.
Sargent’s improvised flick broke the deadlock before he headed home from a well-weighted Milot Rashica cross for the Canaries’ second of the match.
Emmanuel Dennis was shown a second yellow card soon after for a high foot on Max Aarons, before Norwich added a third in added time when Juraj Kucka diverted a ball from Adam Idah past Daniel Bachmann and into his own net.
“No I don’t go away, I want to continue because I am a fighter,” Ranieri said when asked about his future.
“I never give up in my career, this is a good team but they must change the mentality, some players have to follow (Moussa) Sissoko and (Tom) Cleverley, the two captains, this is my word.”
The result ensures the Hornets remain without a win since their shock 4-1 triumph over Manchester United in November, but they have only now slipped into the relegation zone for the first time this season.
The club, though, do not have a history of sticking with under-pressure managers and it could be a crucial week for Ranieri, who described the match beforehand as a “cup final”, although afterwards was left bemoaning the lack of team spirit.
The Italian manager added: “First of all, I want to say ‘sorry’ to our fans for the defeat.
“I asked the players to light their fire and the first half was so not good, not what I expect and the second half, after the goal we disappear and maybe we played better with 10 than 11.
“Now my goal is to pick up the players who pick up with heart, as Sissoko and Cleverley, players with heart, the right mentality Premier League is a tough league and only those who want to fight and play for the squad can continue in this way.”
He added: “The players don’t play for me they play for the club, for the fans for everybody, but the most important thing is they have to play for the squad – not in an individual way, not selfish. I don’t want these kind of players.”
The result leaves Norwich two points above the relegation zone, having climbed above their opponents and earned the praise of manager Dean Smith.
“I mean, it’s great to get the back-to-back wins in this league. It’s hard enough as a newly-promoted team to win any game, but we’ve got back-to-back wins,” Smith said.
“We deserved the one against Everton. I thought today’s was in the balance until the first goal, but it will give everyone a lift.
“That first goal, when it went in, great improvisation from Josh, but I just felt there was only going to be one winner from there.
“I thought there was a calm that came over the players there.”