England lost the bronze-medal match in Nice 2-1, conceding twice in the opening 22 minutes before Fran Kirby reduced the deficit just after the half-hour mark.
Ellen White thought she had equalised moments later but her strike was ruled out for handball following a VAR review.
Phil Neville’s side had been beaten 2-1 by the United States in their semi-final four days earlier. Sweden suffered a 1-0 extra-time loss against Holland on Wednesday.
England right-back Bronze, who won the Silver Ball for the tournament’s second best player, said: “The disappointment of not making the final was enough for us.
“(Saturday’s) game we wanted to win ultimately. But it clearly meant more to Sweden than it did to us, the way they celebrated. We’re a team that wants to be in finals and that disappointment hit us hard.”
England finished third at the 2015 tournament – their best Women’s World Cup performance – via a 1-0 extra-time win against Germany, and were then beaten semi-finalists at Euro 2017.
Sweden, who have had a second and two third-placed finishes prior to this tournament, made the last 16 in 2015, were Euro 2017 quarter-finalists, and were Olympic silver-medallists in between.
Bronze added: “I think we were more disappointed not to make the final than Sweden. That’s what it looked like.
“I know from speaking to some of the German girls four years ago what it was like for them, when they played us.
“We wanted to win the bronze because it meant something to us, it meant history and I think that’s what it meant to Sweden (on Saturday). The disappointment of not making the final hit us hard.”
On winning the Silver Ball, behind American Megan Rapinoe who took the best player prize, Bronze added on Twitter: “Very humbled to receive a personal award, but really not the trophy I wanted to take home. Proud to be able to represent the Lionesses and know this wouldn’t be possible without all my team-mates. This ones for them, and towards our future. Onwards and upwards.”
Neville described the play-off as a “nonsense match” after the game, before stressing that he was “not disrespecting anyone” and saying he had made the remark because “we came to this tournament to win gold.”
Sweden scored in the 11th minute when Kosovare Asllani punished an Alex Greenwood error, and Sofia Jakobsson then fired past Carly Telford.
Kirby replied with a left-footed finish before White had what would have been her seventh goal of the tournament – moving her one clear of US forward Alex Morgan at the top of the scoring charts – disallowed. England were the better team in the second half but could not draw level, with Bronze going close late on when her shot was superbly headed off the line by Nilla Fischer.
White was judged to have been guilty of handball as she jostled with Linda Sembrant before slotting in, referee Anastasia Pustovoytova making the call after watching footage back pitchside.
The handball law was adjusted in rule changes that came into effect on June 1. White, who went to speak to Pustovoytova after the decision, said: “I would love that rule to give the benefit of the doubt to the striker, it would be lovely.
“I’ve got contact, she’s shoved me, I don’t know what I’m supposed to do. I don’t feel like it touched my arm really. I’ve done everything to try to score the goal.
“She (Pustovoytova) didn’t really explain it to me, which I was frustrated with. But she’s there to make the decision, the decision was handball, so I have to take it on the chin.
“She said it hit my arm. And I was like ‘I don’t know what... I don’t know how...’ She just didn’t explain it. I wanted just to get a bit of clarification on why it was a decision to disallow the goal.
“It happened so quickly – I don’t know (if it touched her arm or not). If VAR is here to stay then you just have to take it. But a bit of clarity would be great.”