Does Heathcliff really love Catherine?

Heathcliff does indeed love Catherine. She is his soulmate, united to him in eternity. Theirs is no mere earthly love; in true Romantic fashion, it is positively transcendent, soaring high above the petty restrictions placed upon the development of the emotions by human society.
Numerous examples from the text can be adduced in support of Heathcliff's emotional cruelty towards Catherine, but none of them undermine the central plank of our argument; Heathcliff's love for Catherine is on a different plane and therefore cannot be judged according to conventional standards.
This isn't meant to excuse Heathcliff's behavior towards Catherine, much of which is appalling. It does, however, place his amorous feelings for her in a much wider context. As this is a spiritual (as opposed to a physical or conventional) love, its ultimate veracity is best judged on purely spiritual grounds. That being so, we can adduce Heathcliff's impassioned plea to be haunted by Catherine's ghost as fairly strong evidence of the sincerity of his undying love for her.

Heathcliff loved Catherine from a tender age. He was seven years old when Mr. Earnshaw adopted him. Heathcliff became fond of Catherine Earnshaw as soon as they met, and they became inseparable. The narrator states that Heathcliff’s loyalty was first to Catherine and then to Mr. Earnshaw. This shows that Heathcliff loved Catherine more than anyone else. When Catherine befriends Edgar Linton, Heathcliff becomes jealous of their friendship. This is yet another indication that Heathcliff deeply loved Catherine and could not fathom sharing her attention with anyone else.
Later, Heathcliff escapes from Wuthering Heights after overhearing a conversation where Catherine was telling Nelly that she would not marry him because “it would degrade” her. Heathcliff comes back to Wuthering Heights after Catherine is married to Edgar for five months. This clearly shows that Heathcliff was deeply in love with Catherine and would not stand the thought of her getting married to someone else. Thus, it is in order to state that Healthcliff loved Catherine and showed his affection for her in various instances.