Can the root system or bottom end of escarole start to grow as celery?

Yes! I root my ends in water then transplant to a pot. And set out in garden if you wish when the time is right.

I have rooted the root end of celery in water, but have never gotten past the root stage before the base starts to rot and then it dies before I can get it potted or planted.

Escarole are a member of the endive greens. Celery is different. It is possible you can get the roots to grow on the base like you can celery if that is what you mean. It is worth a try if you want to grow some.

Escarole is a slightly bitter leafy member of the endive family. Celery is a totally different plant.