The Lionesses, beaten by holders the United States in the semi-finals, take on Sweden in the third-place play-off in Nice on Saturday.
Four years ago they collected bronze medals after beating Germany by 1-0 to finish third at the 2015 World Cup.
"We want to win that bronze medal and go away from this tournament holding something," White told.
England have won five of their six games during the month-long tournament in France.
"I think it would be a huge disappointment [if we didn't win] and we hold ourselves accountable for not getting to that final," added White, who has scored in all but one of England's six games.
"We obviously want to win that bronze medal. There's a lot of players that were in the team that got a bronze medal [in 2015] and a lot of players that weren't.
"A medal would be something that we'll be really proud of and we can look each other in the eye and think we did achieve something at this World Cup.
"It may not be gold but a bronze medal is still an achievement.
"Not many people can say they've got a medal at a World Cup and that's our priority."
The final game will be played between The Netherlands and America this Sunday.