Carini stopped her fight against Algeria's Imane Khelif after receiving two strong punches to the face, refused to shake hands with her opponent and then collapsed to the canvas sobbing.
But the bout sparked controversy as Khelif had been disqualified by the IBA during last year's world championship after failing unspecified gender eligibility tests.
There is no suggestion that Khelif, who has fought on the women's circuit for years, including in the Tokyo Olympics, identifies as anything other than a woman.
The International Olympic Committee (IOC) cleared her to fight at the Paris Games as they use the gender stated on the passport as their criterion for eligibility.
Khelif and Taiwan's Lin Yu-ting appear to be caught in the crossfire of a row between the IBA and the IOC, which ejected the boxing body from the Olympic movement after financial and ethical irregularities.
Referring to Carini's reaction at the end of the bout, IBA President Umar Kremlev said: "I couldn't look at her tears."
"I am not indifferent to such situations, and I can assure that we will protect each boxer. I do not understand why they kill women's boxing," added Kremlev.
"Only eligible athletes should compete in the ring for the sake of safety."
The IBA said it would reward Carini prize money "as if she were an Olympic champion."
The association said it was awarding all gold medallists $100,000, of which $25,000 will go to the national federation and $25,000 to the coach.
It was not immediately clear how much Carini would be offered and the IBA did not immediately respond to a request for comment.
Carini has since apologised for not congratulating her opponent.
"All the controversy made me sad and I'm sorry for my opponent too. It's nothing to do with her. She was here like me to fight," she told the Gazzetta dello Sport.
"I was angry because my Olympics had just gone up in smoke but I've nothing against Imane Khelif. On the contrary, if I saw her again, I'd give her a hug," she added.
Khelif fights later Saturday in the women's 66kg quarter-final against Hungary's Anna Luca Hamori, with victory guaranteeing her medal, as all semi-finalists get at least bronze.
