The key fact is that Hawkins's failure to pay the doctor is a breach; Hawkins promised to pay, and he did not.
Many non-lawyers think that a breach by one party excuses the other party from keeping the latter's contractual promise. This is in general not true; a breach excuses the other party only in special circumstances (the tutorials on excuse of performance address this issue).
Assume the doctor sues Hawkins for breach of contract. The doctor's expectation measure award will be:
The fee for the operation minus an amount for the embarrassment the hand causes Hawkins.