"I'm so in love!!!!" she wrote alongside the pic, which shows her gently touching the newborn's teeny tiny bare feet.
The 37-year-old star of The Leftovers also shared a photo of David standing at the entrance to a buybuy BABY store with a shopping bag.
"Who's your daddy!!!" she said. "@davidgardner76 beautiful father best friend Thank you for making me smile so big and laugh so hard and loving me so much."
The Leftovers actress gave birth to Sailor on Feb. 11. She and her beau revealed his name via an Instagram pic of the baby's hand, semi-clutching a tiny anchor pendant. David had posted the first image of the child, which shows him holding onto his dad's finger, on Valentine's day.
Liv also has a 10-year-old son, Milo William Langdon, with ex-husband and Spacehog rocker Royston Langdon. David shares a son with his ex, British actress Davinia Taylor.
In February, Liv was spotted wearing a ring on her ring finger, while a source told E! News that her beau appeared to be wearing one as well. The two have not commented.
Baby Sailor is the second grandchild for Liv's father, Aerosmith frontman Steven Tyler. The 66-year-old rocker told E! News months before the child was born that he is "over the moon that we are adding another member to our wolf pack."