In spite of the superior nature of Sheldon Cooper who always shows his superiority to his room mate Leonard, he cannot continue without him, especially when Leonard was going to shift with Penny and Sheldon left the town. So is there a feeling of care between the two?
There is a recurring theme in the show in which there are moments that show Sheldon genuinely cares about Leonard - all selfish reasons aside.
For example, in this clip towards the end, you can see one such moment: Sheldon talks to Penny when she is contemplating breaking up with Leonard.
Sheldon: Please don't hurt my friend.
Penny: That is the last thing I want to do.
Sheldon: Thank you.
Even though Sheldon has a very high IQ, his EQ or emotional quotient is very much like a 10 year old's and he is shown to care for his friends and Leonard in a similar way (and with a similar reasoning) that a 10 year old child would.
In Episode 515, The Friendship Contraction, we can see how Sheldon genuinely values his friendship with Leonard, beyond that of a mere chauffeur or delivery-man. When Leonard invokes a clause of The Friendship Agreement that allows him to be only an acquaintance, Sheldon concocts a scheme where he turns off the power to the building, then tries to sell Leonard on various activities to do while they wait for the power to be turned back on...but only if Leonard agrees to reinstate The Friendship Agreement. Yes, it's juvenile and ridiculous, but Sheldon doesn't articulate his emotions, and instead prefers to act upon them in a round-about way.
Here's a more recent example from S9E10 "The Earworm Reverberation", as Sheldon records messages for his future self, whom he presumes to be insane due to an earworm:
Sheldon: This is Leonard. He’s your best friend in the world.
Leonard: All right, just stop. This is ridiculous.
Sheldon: Sometimes he gets cranky, but you can trust him with your life. And he does more things for you than I can list.
Leonard: Thank you.