Future predictions are based on models, the effect off co2 on climate is mostly based on empiric data about the ice sheets and co2 levels in the past.
This coupled with models of the sun activity (also from empiric data and physics) results in a pretty strong relationship between these.

I do know a lot about the problems of peer review and not repeatable experiments. I mean, the reviewers can't simply try to repeat all the experiments before approving it, they simply don't have the time for that, they have to trust the data presented, in most cases its a question of reputation. Like, a professor at a university has a lot to lose when providing invalid data. He might lose his job easily if it comes out.

But not only that, we also have to rely on a system where other scientists after you published your study will try to repeat your experiments to try to "proof" your claims wrong.

I read many scientific studies in fields of my interest outside of computer science (as you said, Computer Science as Maths and big parts of Physics are easier since most things can be proven quite easily) and quite often I read several papers where they tried to repeat the data of another study and were not able to.

A possible way to fix the current system is by encouraging people to do so more often by rewarding this more strongly.