Raducanu had promisingly broken in the first game of the third set and then saved a break point on her way to going 2-0 up.
However, Badosa, the world number 62, worked another break point in the sixth game and the Briton succumbed with a double fault.
Raducanu, who is 89th in the world, failed to take three break points in the next game and, on the back of that, lost her serve to fall 5-3 behind.
With Badosa serving for the match, the 21-year-old found the line with a backhand to save a match point as she broke her opponent.
It was only a temporary reprieve as Raducanu let a 40-15 advantage on her serve slip in the next game and, facing match point, served a double fault.
Raducanu was out for eight months – during which time she had operations on both of her wrists and her left ankle – before returning to the tour in January.
She turned down the chance to play for Great Britain at the Olympics because she did not want to switch back to playing on clay at Roland Garros.
Instead, Raducanu has prioritised hard courts the forthcoming US Open – the tournament she won in 2021 as a qualifier.
