Why didn't Naruto's arm grow by itself like Madara or Kaguya? Naruto recovered from a pretty big hole in his shoulder while fighting with Sasuke. Madara was cut in half and could still fly, and Naruto flies, so it would stand to reason that he can recover at least somewhat like Madara. Not to mention how he kind of copied Kakashi's eye, saved Guy-sensei, and tried to save Obito even just before he died.

If Naruto has mastered Senjutsu like Madara and Kaguya, and he has aptitude for recovery, why didn't his arm grow by itself?
Naruto lost his Sage of Six Paths (SOSP) power after his fight with Sasuke. It was mainly due to the fact that he lost the chakras of all the bijuus (tailed beasts) that he had. So, as a result of losing the power of SOSP, he also lost the ability to heal.

Naruto actually 'lost' his SOSP after using a tremendous amount of chakra to seal away kaguya. So after the fight with Sasuke, he must almost have depleted it.

Coming back to the point, while fighting Madara Uchiha and Kaguya, he still had his powers and abilities (levitation, the truth orbs, healing, etc.) It was when he actually sealed Kaguya with the help of Sasuke, did he use up his SOSP powers/chakra. So, he lost the truth orbs, the SOSP cloak, ability to heal, etc.

Now, why did Sasuke not lose his rinnegan? That was because, Indra's chakra is needed to only awaken the rinnegan, and not to maintain it.   

So, the reason why Naruto could not heal his own arm was because he lost his
powers when he sealed Kaguya, but could use levitation , etc. before sealing Kaguya, that is, while facing her and Madara.