Final answer:
The correct completion for the sentence is "died of," which is the appropriate phrase for indicating the cause of death. The sentence will read: "The doctors did all they could, but he finally died of cancer."
Step-by-step explanation:
The correct phrase to complete the sentence "The doctors did all they could, but he finally ________ cancer." is A. died of. This is because 'died of' is used to indicate the cause of death in English. Therefore, the completed sentence should read: "The doctors did all they could, but he finally died of cancer."
The other options, 'asked for', 'fell back on', and 'came into' are not appropriate in this context. 'Asked for' would imply making a request, 'fell back on' means to revert to something as a solution when other options have failed, and 'came into' can mean to inherit something or to enter a place, none of which correctly describe the situation of succumbing to an illness like cancer.