Absolutely he airline should make an offer – it is the airline that is asking for help. If the situation looks serious, and there is no response to a call for doctor, then the airline will almost certainly have to action an emergency diversion.
A doctor’s assessment of the situations seriousness; and then on going management can, as noted by others, save the airline a massive cost and inconvenience. At the latter part of a very long haul, like a trans Pacific, any diversion also runs the risk of pushing crew past duty time, which then becomes very expensive for the airline to deal with (has happened to a colleague on a LAX-SYD that ended up in Fiji. Crew could not continue to SYD, it took two days of pax in hotels to sort out)
Regardless of a diversion or not, the doctor is from that point on effectively ‘on-duty’ until handover to ground medics; and ‘off-duty’ from being a passenger, and can no longer enjoy their paid-for in-flight experience. The airline has in effect hired the doctor to manage the situation whilst in-flight. For sure the doctor could go back to his/her seat and order a series of cabernet sauvignons, but the airline (and the passenger) would be expecting the doctor to attend as appropriate.
An upgrade, or replacement flight (subject to availability) costs the airline very little and would be the least they could do to compensate for taking responsibility and the loss of enjoyment of their flight.