Hi OP,
I believe you are mistaking the
Baby Care Centers for a childcare center for a "Children's Activity Center" in WDW terms.
Here is information on the
Baby Care Centers at WDW:
https://disneyworld.disney.go.com/guest-services/baby-care-centers/
The Baby Care Centers are in all of the parks. These are places you can take a baby for a cool place to rest, to change a diaper or clothes, or to feed your baby (that includes nursing). These are very nice areas - I consistently hear that the Baby Care Center in Animal Kingdom is awesome.
But no - you cannot leave your baby there. It is a place for you to utilize for your convenience to help facilitate your care of your baby while in the parks.
Children's Activity Centers
If you are interested in actual
childcare, there are a few places you could look into. The first and most commonly used are termed
Children's Activity Centers by WDW.
Your choices for Children's Activity Centers would be:
Lilo's Playhouse at the Poly:
https://disneyworld.disney.go.com/recreation/lilos-playhouse/
Simba's Cubhouse at AKL:
https://disneyworld.disney.go.com/recreation/animal-kingdom-lodge/simba-cubhouse/
Sandcastle Club at Beach Club Resort:
https://disneyworld.disney.go.com/recreation/beach-club-resort/sandcastle-club/
Cub's Den at Wilderness Lodge:
https://disneyworld.disney.go.com/recreation/cubs-den/
Camp Dolphin at the WDW Dolphin Hotel:
https://disneyworld.disney.go.com/recreation/dolphin-hotel/camp-dolphin/
You can also look into In-Room Babysitting as an alternative to using the Children's Activity Centers.
In-Room Babysitting
You can also inquire as to having a personal sitter come to your room or even with you to the parks, but you must book these sitters in advance. I believe the advance notice on this is several days to weeks in advance, and you have to pay them for minimal blocks of time - even if you only plan to use them for an hour or two. Here is some more information...
https://disneyworld.disney.go.com/guest-services/child-care-services/
Independent childcare provider Kid’s Nite Out offers one-to-one babysitting in the comfort of your Walt Disney World Resort hotel room. Arts and crafts, reading and playing games are among the fun-filled activities for children 6 months to 12 years of age.
Rates
Number of Children Price (per hour)
1 $16
2 $18.50
3 $21
4 $23.50
After 9:00 PM, a $2 charge will be added per hour. Additionally, a one-time transportation fee of $10 is also required. Please note that rates are subject to change without notice.
In-Park Assistance
You can also reserve a childcare specialist to accompany you and your family during your visit to the parks, providing a helping hand where needed.
Reservations
To contact Kid's Nite Out, please call (407) 828-0920, 8:00 AM to 9:00 PM, 7 days a week. Guests under 18 years of age must have parent or guardian permission to call.