If we want our feet checked, we go to a podiatrist, if we want our hearts checked, we go to a cardiologist. Why, then, if we have a hand problem, do we go to a hand surgeon or a hand doctor and not a manologist?
Well, a word "chiriatry" with the meaning "medical treatment of the hand or hands" supposedly exists (English Word Information). But this word seems to be rare enough that it doesn't actually have a single, specific meaning that is widely recognized—that web page says it can also mean "Healing by the laying on of hands", and The Text Book of Chiropody (by Maurice J. Lewi, 1914) says "'chiriatry' is the only correct scientific expression for manicuring".
As Billy indicated in a comment, podiatrist and cardiologist are based on Greek, not Latin: the Latin roots for "foot" and "heart" are ped- and cord-. The Greek root for "hand" is χειρ-, but it's common to use Latinized forms in the composition of English words, so usually it shows up as chir- (as in the word chiral); occasionally variant transliterations like cheir- and cher- are used. The Modern Greek word χέρι is related, but modern Greek is traditionally not the source of English classical compounds.
The words chirology and chirologist also exist, but don't refer to medicine but rather palm-reading, apparently. "Cherology/cheirology" has been used in reference to sign languages.