The 13-year-old cousin of Ross' was abused during the Dragons loss to the Scarlets.
The Wales back row says fans turned on his cousin after she asked them not to abuse him.
On New Year's Day, Moriarty was shown a yellow card for pulling Sione's head guard.
The number eight was sent off for punching another person.
Paul was an assistant coach at the Scarlets for six seasons.
He wrote that he doesn't mind the abuse he gets online or on a rugby pitch from fans.
My little cousin, who is 13, had grown men screaming in her face for asking them to stop abusing her, that does not sit right with me or should not sit at any sports ground.
I hope people don't think it's appropriate to cross that line.
The alleged incident will be investigated by the Scarlets.
The club said in a statement: "We work extremely hard with our supporters' organizations to create a family-friendly environment for home and away fans and to hear of a young girl receiving verbal abuse from a fan is completely unacceptable."
Fans are encouraged to report instances of abuse at the stadium to the club.
We will investigate this further if the family would like to contact us.
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