Yes, medico is dative. (Your understanding of it was right all along, it seems like you just confused the names "ablative" and "dative.") And to explain it in a slightly different way, since the verb ostendo means 'to show' or 'to hold/stick out', by its very nature it will usually be accompanied by a verb in the dative, expressing 'to whom' the thing is shown or held out to.
As Quickly was saying, you can't have potest because it is indirect discourse. What Aemelia thinks (expressed directly) is, "Medicus puerum aegrum sanare non potest." When you turn it into indirect discourse, introduced by puto, it requires both the subject and object to be accusative, and the verb to be an infinitive (the so-called "accusative-infinitive construction," which is very common).
Hope that helps!
Dic mihi, Damoeta, 'cuium pecus' anne Latinum?