Can a jewelry store steal your diamond out of your ring when they say there going to clean it?

Yes.

And in order to prove it, you'll need a certificate which documents the stone. Every diamond stone is individual and unique given its cut, colour, clarity and carat weight.

You can pay a certified gemologist to document your diamond. It's a valuable document to have, because any insurance claim or in you case, claim for theft will be based on this evidence.