Sophie Price, 22, started the petition after her five-day-old daughter, Florence, was admitted to intensive care at The Grange University Hospital in Cwmbran.
Ms Price said she was refused a meal because she was formula feeding while Florence received breathing support and critical care.
The petition calls for NHS trusts across the UK to provide meals to mothers whose babies are receiving hospital care, regardless of whether they breastfeed or formula feed.
Ms Price said: “I’m overwhelmed with the support I’ve gained since starting this petition to make sure all mothers are treated fairly up the hospital.
“Whether they breast feed or formula feed, they should be provided food when they can not leave their five-day old baby who’s on intense breathing support.
“The petition hitting over 1,000 signatures as fast as it did is a testament to how many other parents feel as strongly about this as I do.
“I’ve received countless messages from women who have experienced the same as me, being refused food all because they had formula fed their child.
“These women all have different reasons why they couldn’t breastfeed, such as cancer, or medication that could harm the baby, low milk supply and so many other reasons, which really opened my eyes to why this petition is ever so important to make a positive change for women for generations to come”.
Supporters of the petition have also shared their own experiences with Ms Price.
One woman said: “As someone who never got a milk supply after a complicated birth, the current approach is grossly unfair.
“My choice was taken away, yet this would be a double punishment.
“I don’t understand the ‘why’ – do they think withholding support will encourage more to breastfed?”.
Another wrote: “My baby was born this year and was on a ventilator.
“One thing that made life so much easier was the free healthy meals provided to me – this should be given to every parent! Should not be a postcode lottery”.
Ms Price and baby Florence appeared on ITV’s Good Morning Britain on Thursday, October 22, to discuss the petition.
She thanked people who had signed and shared the petition, saying she hoped to take the campaign as far as possible for women unable to speak up for themselves.
While the petition continues to gain momentum, Ms Price has continued to praise staff at The Grange University Hospital for caring for Florence.
She said: “The care my daughter received in The Grange University Hospital was phenomenal, and I’m really thankful for all the staff who worked tirelessly to keep my daughter alive and breathing with her being so ill”.
The petition is seeking support for a UK-wide approach to meals for mothers in this situation.