it rather depends on what you mean by 'prove'. Certainly in everyday usage statistics provides proof - for instance a paternity suit might be settled to everyone's satisfaction with a DNA test.
The word "prove" is often loosely used. For example the existence of the Higgs Boson (or more properly, the Higgs Field) was "proved" to scientific satisfaction by statistical tests - but all the scientists understand that. They also know that further research might lead to different conclusions, as has often happened in the past (eg phlogiston).
Statistics suggested a link between smoking and lung cancer, but it was only proved by showing clinically that carcinogens in the inhaled product could cause cancer in the lungs. (Of course the actual incidence of cancer is still down to chance - some people get away with it!).
Too many people take "prove" too literally. Some have claimed that a particularly striking result has "proved" that astrology is true. They failed to understand that if you do a million trials, you actually expect a 1 in a million event to occur.
Over the years I have spent far too much time dealing with misunderstandings of this sort.