Yes, you can prove a theory

You just have to be very careful about how you define "proof" and "theory". Which sounds silly, but it isn't really. If you accept that reality is an illusion, and/or that our senses cannot be trusted or give information which is always unreliable, that we cannot ever really know what reality is like, then it's true you can't prove a theory. But in that case you can't even have true facts either, only things which (at best) can be assigned probabilities.

But this is not typically how most of us think or how science normally works either. It operates under the assumption that we can actually objectively measure the world, even if our information about it is not complete. Our measurements do represent reality in some way. They may occasionally be wrong or people may lie about them and deceive us, but this is the exception rather than the rule. Thus, under this assumption, it is perfectly possible to measure things we may consider to be absolute, true facts. The only way to say the Earth is flat is to invoke a massive conspiracy theory and/or our memories are being constantly manipulated. This isn't wrong, but it isn't science.

Once you allow absolute facts, you also allow provable theories. Evolution has been proven to occur, the Earth has been proven to be roughly spherical. These are both theories, in that they tell us how the world works, and facts, in that they are known to be true. They have varying degrees of predictive power, and you have to specify precisely under what conditions they should occur - but if those conditions are indeed replicated precisely, then evolution will always happen and gravity will always operate.

It's true that most of the time things are not always so black and white - most theories are varying shades of grey. But the absolute extremes - the definitely true and the definitely false - do happen. "Alternative facts" are indeed simply lies.

With cast of mice, necromancy, snakes, cats, and a tiny elephant... see also this post about proving negatives.

You Can't Not Prove It Wasn't Me Who Didn't Do It

Science is about working out how the world actually functions. It doesn't have any truck with hocus-pocus mumbo-jumbo or fairy stories about some magical deity running the show - it's about solid, hard facts, evidence and proof. Nice, comfortable, iron-clad certainties all the way !