Unfortunately we don't have a REST resource for this in the OpenMRS reference application.
(The Bahmni distro has added one here, and perhaps someone could port this into the emrapi module.)
What you'd have to do to get this from the standard reference application is to get from the
concept resource for the given patient and the "Visit Diagnoses" concept (uuid=159947AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA), and parse these obs groups.
PS- If someone creates a ticket for adding a "diagnosis" REST resource via the emrapi module we should definitely mark this as a community priority.