To echo what cailindana said, when I became a mother, one of the hardest things I found to deal with was the fact that in the end, it came down to me to settle the baby. That I had to just suck it up and cope. There was no other option. DH was and is a good dad, but the times when only mum will do can be draining. Father's usually have the option, particularly in the early days, of passing the buck to mum.
I don't really want to describe myself as 'better' than DH at childcare, because I don't feel I am a better parent, but more often than not, I can settle the children quicker. But that isn't because I'm better, it's because it's me that's always done it. Maybe it would be different had our roles been reversed. If it had been him primarily meeting their needs, they'd probably ask for him first over me.
I think the 'nurturing' role goes beyond childcare too, such as when caring for elderly parents or other adults. Women are more likely to be providing unpaid care than men, and are more likely to reduce paid work in order to do this (which can also end up with them having smaller pensions).
Anyway, to actually answer your question OP, I think society in general accepts/assumes that women are better than men at childcare. But I'm not sure how this joins up with the attitude that fathers that take on the primary childcare role are somehow better, as has been mentioned on this thread. I agree with others here that if women are 'better' than men, it's only because women do the most of it.