Is it the same child that keeps pushing? If it is, I would consider talking to a lawyer. I say this because there are times when a daycare NEEDS to let go of a child for the sake of the other children. It's hard to let go of revenue for something that seems like a normal childhood problem. All kids push sometimes. But, sometimes a child does something like that again and again. So when does a provider draw the line with the one child? All kids will do the same behavior several times before they finally stop. Accidents happen VERY fast. And yet, I haven't had one single accident like this, this bad, in 24 years.
However, one time a child clobbered another child with a toy and that child ended up needing staples in the top of his head. I felt horrible. These boys are BEST friends! The mother of the hurt child wasn't even angry. She knows me and she has known of the other child for years. So all was good. But imagine how different it might have been if it was a new parent and they didn't know how careful I am?
The entire reason to sue a business is to make an example of them and or to recover money lost. They should be paying your co-payments assuming you have insurance. This really could have happened while in your care. Through the years, my kids have come in Monday morning with lumps, bumps, casts, gashes, stitches, stories of stomachs being pumped and charcoal being administered in the ER. All of these things have happend at home with their parents.
If you don't believe they are being negligent then there should be no lawsuit. If someone is talking on their cell phone and not doing their job, then you have a case. If a child is mean and has done this sort of thing repeatedly and they were not separated from the group or let go, then you have a case. If these are random kids doing it randomly, then you need to find out how they are handling discipline because they obviously do have a problem that needs to be addressed and fast.
There are many ambulance chasing lawyers that will take your case. Just be sure you are doing it for the right reasons.