One post shows the woman raising her middle finger toward the newborns in the hospital with the caption: "How I currently feel about these mini Satans."
A video shows her picking up one newborn and moving his or her arms as if the baby is dancing as music plays in the background.
The posts have been shared tens of thousands of times, with many online commenters condemning the actions as insensitive and outrageous. Many people have also contacted the hospital directly to express their anger.
Naval Hospital Jacksonville officials said they are aware of the video and condemned the posts as "outrageous, unacceptable, incredibly unprofessional."
Some of the initial online posters identified the individuals involved as hospital nurses. The hospital, however, said the two individuals were not nurses, but are junior enlisted corpsmen.
"They have been removed from providing patient care and they will be handled by the legal system and military justice," the hospital said in a Facebook statement.
The parents of the infants involved have also been notified, the hospital said.
The woman later told a friend her own actions were "childish" and were done out of boredom. She apologized to the friend for her actions, according to ActionNews Jax.