The series stars Ellen Barkin, Justin Bartha, Jayson Blair and NeNe Leakes, focusing on a gay couple who hire a surrogate mother.
Representatives from One Million Moms posted a statement on their website, criticising the show for its inclusion of homosexual characters.
The statement reads: "NBC is using public airwaves to continue to subject families to the decay of morals and values, and the sanctity of marriage in attempting to redefine marriage.
"These things are harmful to our society, and this programme is damaging to our culture."
According to Zap2it, the group is threatening to call the show's sponsors in a bid to get it off the air.
However NBC has yet to respond to the group's demands, and The New Normal is still set to premiere in the fall.
Watch a trailer for The New Normal below: