No, not as such, but there are a variety of options open to the Captain depending on the nature of the incident and whether it can be dealt with in-flight, or requires a diversion to an en-route airport.
Many factors come into play. A doctor would almost certainly be requested to assist if there was one amongst the passengers. First aid would be administered by the crew, or in conjunction with the resources available on board. Advice by radio would be requested from a medical contractor, who would be able to better advise, in liason with their own team of 24 hour on call doctors. Discussion would also take place between the Captain and the first aiders, and or cabin crew, to better form a consensus on which a decision can be made.
Based on the advice and culmination of all the information available, a diversion might be made to a nearby suitable airport. Clearly the safety of all the passengers and crew is always of absolute priority and in some circumstances that might mean that the medical emergency has to surrender to those priorities. The nearest suitable airport might be nowhere near suitable medical facilities, or the weather or other operational factors might preclude the nearest choice.
In the case of a diversion, the equivalent of a port health authority may be requested to attend, and in any event the procedures followed would be those of the national authority of the country concerned. In an emergency there would normally be an expected level of assistance and co-operation.
Although it is something of a moot point, the jurisdiction applicable to an aircraft once power has been applied for the take-off roll is that of the country of registry of the operator (which may differ from simply the country of registry.) Once the aircraft has landed, jurisdiction subverts to the host nation, albeit there may be international treaties that modify standard rules within that jurisdiction.
As far as the FAA and CAA are concerned, they would expect the crew to operate within the normal rules and regulations pertinent to air transport operations within the normal jurisdiction. But in either case, would devolve the final authority for any variances to the Commander, in the case of an emergency.
As for the biting insect, I would suggest an insecticide spray or a rolled up copy of the Sun. Naturally avoiding any allusion to the suggestion of a pair of ticks on page 3!
I hope that has answered your question. Each situation is different, and therefore there is no universal correct answer. It would be a judgment call that takes into account the nature of the emergency, the location and the available resources, in any number of possible combinations.