A separate statement from the Dax diocese said the resignation was over the bishop's behaviour with young people.
"For several weeks in the Dax diocese, rumours persisted over the bishop's inappropriate pastoral behaviour," the church statement said, adding that that they had been brought to the attention of church authorities.
Pope Francis accepted Gaschignard's resignation, it added.
A spokesman for the diocese, Paul Perromat, told AFP the resignation followed "possibly inappropriate comments and behaviour" by the bishop but that "it is in no case a question of sexual aggression or acts."
He said: "As of now, no complaint has been lodged" with local prosecutors, adding that the church "thinks of all those hurt in this matter".
Ricard said two church workers "in contact with youths" had approached him on March 21 to voice their concerns.
Gaschignard's "behaviour upset and troubled a certain number of these young people, who spoke to their families about it," Ricard said.
The Vatican intervened in the case with a directness and rapidity it has often been accused of lacking over other cases, especially with high-ranking clerics accused of abuse or covering up for it.
The French church was rocked last year by allegations that the Archbishop of Lyon, Philippe Barbarin, covered up the abuse of scouts and failed to remove a priest despite being aware he had sexually abused the boys a quarter of a century ago.
A string of historic paedophilia cases in North America and Europe has unleashed widespread criticism of the Catholic hierarchy.
The Dax diocese, to which Gaschignard was named in 2012, was to hold a news briefing on Thursday afternoon.
The Barbarin scandal was the worst to hit the church in France since 2001, when a bishop was given a three-month suspended jail sentence for failing to inform authorities about a paedophile priest.
Barbarin, who has protested his innocence, remains in his post, but the scandal has tarnished the image of one of the church's most media-friendly figures.
In response to the scandal, the church's hierarchy announced a series of measures last April, including the creation of a dedicated secure email address for victims to report abuse.