The "Hero" hit maker gave birth to twins -- a girl named Monroe and a son named Moroccan -- at a Los Angeles medical facility on April 30, the date of the couple's third wedding anniversary.
Cannon reveals the babies are still in hospital -- and he also admits he and his wife have spoken to officials concerned about their tots' welfare.
In an interview with Piers Morgan on Thursday, the rapper/actor said workers from Child Protective Services contacted the couple after they were wrongly informed that the babies were exposed to alcohol.
Explaining the situation to Morgan, Cannon said, "Recently, the Child Protective Services were called with allegations that there was some drinking and drugs and all that going on, while in the hospital, which makes no sense to me. How would a hospital allow that?"
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"It all started when a nurse suggested to my wife that if you drink Guinness, the dark beer, the yeast improves breastfeeding ... I don't know if someone heard that, but then they were saying my wife was drinking beer. People will do anything to try to conjure up a story.
"Then when I spoke to the person from Child Protection Services, [they said], 'This is ridiculous, we're going to make sure this isn't the case.' To even have to deal with that, my wife in the state that she's in, we're in the hospital, to even have to think about someone possibly wanting to investigate your children ... it's sad at the end of the day."
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Cannon also claimed that paparazzi eager to land the first photograph of the twins have been trying to infiltrate the medical facility.
He added, "There's been photographers actually in the hospital posing as different people or trying to get pictures of the kids, which is really sad when you think about people trying to make a buck off newborn babies."
The full interview with Cannon will air on "Piers Morgan Tonight" later this month.