Can a barking dog damage my newborns hearing?

Not likely. It may startle and scare the snot out of the baby though. Pretty sure you don't want that...
Only if close to ear. Very loud noises, such as an explosion, a gun being fired, etc can cause immediate damage in any person, young or old. A dog barking nearby to a child is unlikely to cause hearing damage, unless he barked frequently and constantly very near the newborn's ears.
Barking Hearing Loss. No, a barking dog will most definitely not cause a hearing loss. Noise greater than 90 decibels, and then over a prolonged time period, depending upon just how loud the sound is, is necessary to cause hearing loss. A barking dog just is not loud enough to cause hearing loss.
Hush little baby. Newborns have the best hearing of anyone alive. For the better part of a year, they were under water so, when they're born, our world is incredibly loud to them. Loud noise can damage the organs of hearing so, please protect your baby. Some noise is okay so they get used to us but avoid barks, phones and such. Your baby also should have had a screening hearing test before leaving the hospital.