Bristol's mother, former vice-presidential candidate and Alaska Governor Sarah Palin, spoke to ET exclusively about Meyer's claims.
"For many months we have been trying to reach out to Dakota Myers [sic] and he has wanted nothing to do with either Bristol's pregnancy or the baby," Palin told ET.
"Paramount to the entire Palin family is the health and welfare of Sailor Grace," she added, claiming that Meyer is attempting to "save face."
Meyer filed legal documents on Monday alleging that he's Sailor Grace's father, and is seeking joint custody of the newborn girl, as well as child support.
Bristol, who has never publicly revealed the identity of Sailor's father, got engaged to Meyer in March 2015 and planned to get married two months later in May. One week before the wedding was set to take place, the nuptials were called off.
In June, it was revealed that she was expecting.
Bristol's rep, David Martin, also addressed Meyer's legal filing, telling ET, "My values are such that a real American hero doesn't ask for child support."
Meyer, a Marine Corps veteran and Congressional Medal of Honor recipient, shared a photo of Sailor Grace to Instagram, shortly after she was born on Dec. 23.
Meyer captioned the photo, "Best Christmas present ever!! I couldn't be more proud of this little blessing."
According to Bristol's rep, it's unclear where Meyer received the photo he posted.
ET has reached out to Meyer for comment.