Yes, Heathcliff is nasty, and it does seem odd that Linton would stop writing to Cathy, but several things happened. First of all, don't forget that Linton was very sickly and did not have much energy for anything. Everything tired him out. In chapter 23, when Catherine is told by Heathcliff that she was wrong to have stopped writing to Linton and that he was probably going to die and it was all her fault, she persuades her father and Nelly to let her visit Linton again, and when she sees Linton, Linton tells her that he was, indeed, too tired to keep up the correspondence:
“Why didn't you come before?” he asked. “You should have come, instead of writing. It tired me dreadfully writing those long letters. I'd far rather have talked to you. Now, I can neither bear to talk, nor anything else.
Also, in the prior chapter 22, when Catherine and Nelly run into Heathcliff on the Moor, Heathcliff chastises Cathy for playing with Linton's affections and only "playing at love" with her letters. One gets the idea by his speech to Cathy at this point that after Nelly insisted that Cathy not continue writing to Linton and after she would not show his letters to Cathy, Linton probably started whining about it and then Heathcliff and Hareton began to tease him unmercifully. Heathcliff hated his sickly son and probably gave him so much grief about pining away for Cathy, that perhaps he didn't feel like writing anymore -- and why should he, if she was not going to write back? He did not have the energy. Here is what Heathcliff said:
Two or three months since, were you not in the habit of writing to Linton? making love in play, eh? You deserved, both of you, flogging for that! You especially, the elder; and less sensitive, as it turns out. I've got your letters, and if you give me any pertness I'll send them to your father. I presume you grew weary of the amusement and dropped it, didn't you? Well, you dropped Linton with it into a slough of despond. He was in earnest: in love, really. As true as I live, he's dying for you; breaking his heart at your fickleness: not figuratively, but actually. Though Hareton has made him a standing jest for six weeks, and I have used more serious measures, and attempted to frighten him out of his idiocy, he gets worse daily; and he'll be under the sod before summer, unless you restore him!”
So stopping the letters was for a combination of reasons.