You can't prove that you have a normal distribution by looking at data; there will always be some other, more complicated distribution (an infinite number of them in fact) that are closer to the distribution of some sample than the sample is to a normal distribution.

You can, with enough data, show that you don't have normality. Indeed often you can tell for certain with no data at all (most biological measurements are strictly positive)

As with any distribution we have named, normality is a simple model, an approximation, not a biological fact.