In the hospital rooftop fight between Sasuke and Naruto, why didn't Sasuke use his Sharingan to learn Naruto's Rasengan? Is it because the Rasengan doesn't use any hand signs? (But Sasuke learns Rock Lee's moves with the Sharingan even if it doesn't rely on hand signs).
Sasuke's chakra affinity is Fire and Lightning.
In the anime, it is speculated by Kakashi that the best chakra affinity to combine the Rasengan with is Wind.
With this in mind, it was probably best for Sasuke to focus primarily on electric or fire based attacks, such as Chidori/Raikiri and Katon. He would be able to elevate these jutsus far more than he can with Rasengan.
The Sharingan was mainly used for casting Genjutsu, following an opponent's moves, which could be the hand signs they make, and to imitate the opponent. Sasuke couldn't just "copy" the Rasengan, he would of had to master it too. And if you remember when Naruto was practicing to master the Rasengan he had to go through 3 hard steps. Which were with a water balloon, a rubber ball, and an air balloon.
As I stated earlier the Sharingan can imitate the hand signs of an opponent but the Rasengan did not require any hand signs so it would have been hard for Sasuke to "copy" it if he did not know what to do. Also to master the Rasengan the ninja requires extremely refined chakra control, something that very few ninja naturally possess. That's why the Rasengan is difficult to master. In order to help others gain the necessary chakra control, learning to use the Rasengan is broken into three steps:
- Rotation - the user learns to spin their chakra in multiple directions at once. To help in this regard, users may be given a water balloon so that they can mark their progress by the churning water; this step is complete once the users is able to burst the balloon with their chakra. It is important for the user to know which direction their body naturally spins its chakra for this step.
- Power - the user must increase the volume and density of chakra they output. To help in this regard, users may be given a rubber ball: there is no water inside to help them burst the exterior and the rubber shell is thicker than a balloon's.
- Containment - the user must combine the first two steps into a contained sphere. To help in this regard, users may be given a balloon to help them visualize the intended shape; if the balloon is popped or otherwise moves, mastery is not yet achieved.
I'm not sure if Sasuke has refined chakra control, but if he didn't it would have been really hard to master it, like Naruto.
This is kind of of my opinion but I don't think Sasuke would of wanted to learn something that is hard and takes time to master. He would of wanted something quick but powerful, like the Chidori, to master like that he could get ready to kill Itachi.
Also an addition to your statement in your question Sasuke did not copy Lee's moves he just mimicked them. I know this because Lee uses Taijutsu, something that requires no hand signs nor chakra, and Kakashi one time stated that the Sharingan can NOT copy Taijutsu.
Plot-holes. I love Naruto but if there was ever a real Akatsuki, Kishi would be in it as a S-class plot hole creator. Anyway Sasuke could copy the Sasengan just as Kakashi did; plus he made Boruto learn it so that he can master shape manipulation at it's best. Before thinking why Sasuke didn't go to learn the Rasengan is due to:
- Two contrasting teachers (Jiraiya - work hard type & Orochimaru - genius type)
- Sasuke's god complex (thinking he's right in everything)
- Sasuke intent from the beginning was to kill Itachi and Chidori is the best type for him with his speed and Sharingan.
But if we consider all this criteria and as Boruto learns from Sasuke, we might as well assume that as both has six paths power they (both Naruto and Sasuke) can use Rasengan and Chidori.
Assumption of course but it is plausible considering they have mastered yin/yang styles and five elemental natures.