What's bothering me about this is probably a semantic problem: the difference between an opinion and a belief. Many people believe things for which there is no factual evidence, or which is contrary to the factual evidence that is available. Beliefs (eg that the Earth is flat or that drinking green tea prevents ingrown toenails) like that are neither valid nor invalid, so long as they do no harm to anyone else. In the OP's example, "eating ice-cream cures cancer" is a belief, not an opinion.
An opinion implies an assessment of facts or experience, or may be value-based.
I don't see how, logically, an opinion that is factually incorrect can be valid. "In my opinion, Skakespeare's plays are written in French" is clearly wrong." However, "In my opinion, Shakespeare's plays are trite, poorly-structured, and historically irrelevant", while possibility a minority viewpoint, can be argued with but is not objectively wrong. "I find Shakespeare's plays really boring" is a valid opinion, and can't actually be contested.