Did Rakesh remain a devoted son until his father's last days?

Rakesh tries very hard to be a devoted son to his father, actually prolonging his life by improving his diet and bringing him vitamins and medicines, but he is not, the text suggests, doing the best he could for Varma—which would be to let him die on his own terms.

It is clear that Rakesh is not following his father's wishes by prolonging his life. He is devoted to the idea of keeping his father alive rather than to his father himself. As the narrator states:

The quantities of vitamins and tonics he [Varma] was made to take were not altogether useless. They kept him alive and even gave him a kind of strength that made him hang on long after he ceased to wish to hang on. It was as though he were straining at a rope, trying to break it, and it would not break, it was still strong. He only hurt himself, trying.

What makes the story interesting and complex is the question it raises of what devotion is. Does it mean complying completely with a loved one's wishes, even if those wishes are self-destructive; or does it mean doing what is best for the parent, even if it isn't what he wants?

Rakesh takes the road of doing what he believes is best for his father even though his father would prefer to eat rich, unhealthy food and die earlier. The text, however, implies that this is not the best path and that the son is insensitive to the father's real needs:

"'Lying all day on his pillows, fed every few hours by his daughter-in-law’s own hand, visited by every member of his family daily—and then he says he does not want to live 'like this,'" Rakesh was heard to say, laughing, to someone outside the door.

The laughter shows Rakesh's insensitivity to his father's real needs. Rakesh is not as devoted as could be. It is humiliating to Varma to have his life prolonged and to be unheard. He begs to be let to die, but his words are not attended to. The story ends him with saying that God is calling him and demanding to be released from life, but we don't know what will happen—we can guess, however, that Rakesh will again try to save him against his will.

Is Rakesh truly a devoted son?

The original question had to be edited down.  Though the ending of the story might not present it to be such, I think that Rakesh is a devoted son.  He is devoted to his parents.  He does everything that is asked of him.  He does well in school, achieves and succeeds in a profession that brings his parents pride and joy, marries a woman of their choosing, comes back to India when they ask, and tends to his father when he is a weakened state.  He is devoted in that he is able to put his aside his own wishes for their betterment and their word.  The bond that he carries as their son does not dissipate as he gets older.  In other words, he remains their son, even taking priority over as husband or father.  

In this light, Rakesh would have to be seen as a devoted son.  There is little more he can do.  The ending of the story where he denies his father his wants is only done because of health concerns.  Rakesh is simply using his own knowledge, gained at the wishes of his parents, to help his father's own health.  He is a devoted son because the question remains as to what else he needs to do to make them happy.

Does Rakesh remain devoted until the end of "A Devoted Son"?

I think that one could say that Rakesh did remain devoted to his father and his family until the end.  Rakesh never really changes.  He is shown to be a devoted son at the start of the story and this is how he is at the end.  He acts and embodies the reverential son throughout.  I think that there are two ways in which to see this.  In one respect, Rakesh fulfills the responsibility of children in the traditional expectations of Indian society.  He does his parents' bidding.  They demanded that he respect their word and be obedient.  When Rakesh needed to come back to India, he did.  When he needed to marry a woman not of his own choice, he does.  When he is expected to adhere to what social standards expects of him, he does.  Hence, this makes perfect sense that when he is needed to cure his father, he does not shy from this and pursues it vigorously and with devotion.  In this light, he does represent a "devoted son."

On the other side of the coin, perhaps the ferocity and intensity within which he treats his father is a reflection of all that he has had to sacrifice because of his devotion.  Rakesh is not shown to be one who has a voice at all.  He does his parents' bidding.  Perhaps, it is for this reason that he dominates his father with so much zeal in the hopes of curing him.  It would simply make no sense for Rakesh to have given up his life for his parents and his father, in particular, only to see him die.  It is here where Desai might be showing that the demands placed on children, if they are not monitored to allow some level of voice and acknowledgement, devotion can be brought out in other ways.  The dominating aspect in which Rakesh relates to his father could be more of a reflection of his own dissatisfaction with having lived his life for his parents and the inability to accept the reality of age and death could be his response to such a condition.  It is here where Rakesh's devotion could carry another side to it.  Regardless, I do think that Rakesh's devotion is present in the story.

Is Rakesh consistently a devoted son throughout the story?

It can be argued that Rakesh is a devoted son throughout, but his extreme devotion later results in unintended, negative consequences for his aging father.

We are told that Rakesh has always shown a high degree of filial loyalty to his father, Varma. When the national examination results were announced, Rakesh bowed down and touched his father's feet in reverence. In fact, we are told that this is the first thing Rakesh did after seeing the results in the newspaper.

Not only is Rakesh a devoted son to his father, he also shows a great degree of tender regard for his mother. Even though he studied in America, Rakesh did not bring back a foreign bride to his parents' home. In fact, he did not stray from the Indian tradition of allowing his mother to choose his bride. The text tells us that Rakesh married an uneducated, old-fashioned village girl of his mother's choosing.

It can be seen that Rakesh met all the fond expectations of his parents. Because of Rakesh's filial loyalty, his parents are the envy of the whole village.

Later, however, Rakesh's extreme devotion results in unintended consequences. After his father has a health scare, Rakesh begins to exert stricter control over the older man's diet. He prohibits Varma from having any desserts or fried food. As a result, Varma's quality of life decreases. In the end, the old man begins to contract actual diseases. This leads to Rakesh doubling down on medicines for Varma.

Rakesh is a devoted son throughout, but his devotion blinds him to his father's true needs. In the end, his medical training fails him. He is so focused on treating physical ailments that he forgets to tend to his father's emotional needs.