I wouldn't base your future in child care on this one instance. Did you tell the dcm you were talking with, that the child was having a temper tantrum? It was probably pretty obvious even if you hadn't mentioned anything.
It's incredibly difficult to say if you were in the right/wrong or if your employer was being hot-headed. Maybe she took issue because you weren't tending to the child? I have no clue, having not witnessed it.
If childcare is where you think your heart is, use this experience to your benefit and learn from it(believe me, it's not for everybody and not a simple job despite what many think). Was the employer willing to talk with you about it or more of a 'you're fired, get out' kind of reaction? And why would the other employee be running to tell the employer about it? Did she feel you were at fault? Maybe she thought you were more interested in chatting with the dcm rather than helping the child? Again, have no clue. It sounds like it was a chaotic situation that needed tending to and the dcm should've waited.
I think working in a center, under someone else, has got to be so hard.
Hope everything works out for you.