No, disease names are not proper nouns, although diseases named after people keep the capitalization of the person's name (Münchausen syndrome). The scientific (Latin) names of disease-causing organisms follow the standard rule of Genus species.
If you've seen "Celiac Disease", it's just because of an unfortunate tendency some People have to capitalize Random Nouns as if English were half German.
Diseases are not capitalized unless the name comes from the name of a scientist.For example rinderpest is not capitalized while Marek's disease is capitalized since this name comes from the researcher Marek. Similarly pathogenic organisms eg. anthrax bacteria are not capitalized. Of course, when we are writing the binomial name of the organism ( Bacillus anthracis) the genus name is capitalized. Too many people are writing this casually and young researchers are confused or learning wrong things.