The baby, who is being cared for at Edinburgh Royal Infirmary, has been named Charlotte after the police woman who accompanied her to the hospital.
Baby Charlotte weighs 2.6kg (5.7lbs) and all routine newborn checks have been carried out.
She was discovered by a passer-by in Hailesland Park at 12:55 on Thursday.
Lawrence Liddell, 59, from Hailesland Park, told BBC Scotland how he was still in shock after finding baby Charlotte.
He said: "My friend and I saw a baby lying on the bench all wrapped up, waiting to be found, it looked like, because her blanket had been folded into a hood and tied with a bow.
"It was a total shock and it all feels surreal even now.
"We took her to the concierge, which was the nearest warm place, and called the police.
"From what I could see the baby was well, she was lovingly wrapped and I just hope the mother is ok.
"We looked around for the mum but we could not find her."
Call for 'baby boxes' for abandoned newborns
NHS Lothian also said tests have been carried out on baby Charlotte to rule out infection including Group B Strep.  As a precaution Charlotte is being treated with antibiotics.
Maria Wilson, NHS Lothian's chief midwife, said: "Baby Charlotte has had a good night and is doing well.
"Our staff are continuing to care for her and we hope the mum and baby can be reunited.
"I don't want the mum to feel afraid of contacting us because we treat everything confidentially.
"I think the things we would want to check are that she is not bleeding or she's not got any infection or any problems post the birth of her baby that she needs assistance with."
Baby Charlotte was thought to be no more than a day old when she was found.
Police are now checking CCTV footage from the area where Charlotte was found.
A newborn baby girl who was found abandoned on a street bench in Edinburgh is "doing well", the hospital has confirmed.