Shortly after telling the BBC his story the father, called Jahangir, received a call from Ashraf Ghani assuring him that arrests would be made.
His wife was allegedly raped in Badakhshan province eight years ago.
But Jahangir says he has been unable to get the alleged rapists arrested because they are politically powerful.
His wife told BBC Persian about the incident.
"I told one of them, for God's sake, I have just given birth, I'm like your child, even your daughter is older than me," she said.
"I cried a lot, they beat me all over my head and body," she said.
"Their bite marks were still on my body until a couple of years ago."
Analysis: Inayatulhaq Yasini, BBC Pashto
The stand taken by Jahangir and his wife breaks something of a taboo in Afghanistan, where victims of sexual crime are generally cowed into silence.
Soon after the media picked up the case, President Ghani spoke with Jahangir and promised him that "the case will be investigated properly and perpetrators will face justice".
The victim herself told the BBC: "I want justice to save other women from such a fate."
In the wake of the publicity, the Afghan government will be under pressure to investigate the incident properly and ensure some kind of resolution.
But there are also hopes that the stand taken by Jahangir and his wife will embolden other victims of sexual crime, and have a lasting affect on how such cases are dealt with.
Jahangir, a policeman, told the BBC eight men were arrested after the incident, but six have now been freed because of their political connections.
He says he and his family had to flee to the capital, Kabul, fearing for their lives.
He has threatened to kill himself unless that changes: "If they are not put on trial, I'll have to commit suicide to escape the tensions I'm suffering from. There's no other way."
Jahangir went public with the threat on Afghan television. He later met the country's interior ministry to press his case.
While speaking to the BBC following the meeting, he received a phone call from Mr Ghani. He wept as they spoke.
Mr Ghani was sworn in as Afghan president in September, pledging to tackle the issue of corruption as a priority.
As well as political corruption, activists say violence against women is rife in Afghanistan, but rarely attracts much public attention.
In one case that did spark national outrage, five Afghan men convicted of gang raping four women were hanged in October.
The president of Afghanistan has told an emotional husband that a group of men accused of gang raping his wife would be brought to justice.