And no-one has the right to just believe something without proof.

It would seem that you believe that. So what is your proof?

And that is a moral statement to boot.

From the more physicsy side, I'd guess you believe that the sun will rise tomorrow, and that electrons at the center of the earth have the same rest mass and charge as the insanely small number of electrons that have actually had their mass and charge carefully measured. You probably believe a or not-a and that 2+2=4. Are you familiar with Godel's theorem? My recollection of it is that he proved that any formal system above a certain trivial level of complexity had true statements in it that were unprovable.

I can tell you the easy way out is to claim you don't believe anything, that it is all in your head, not just the moral stuff. Of course then we need to invent new language to describe the difference between statements like "no-one has the right to just believe something without proof" and "no-one has the right to just believe something with proof." I guess you could say that just because you believe something doesn't mean you believe it is true, but that starts to sound more like a non-standard definition of the word "believe" than anything with useful content.

As to my explaining to you the details of why anyone should be a moral realist, I"m not interested in attempting that. I'm not a committed moral realist myself, and I'd just have to do a lot of reading myself to find a good description of what ask for. Sorry.