I understand that Doctor Manhattan was very much out of touch with humanity at the end of the graphic novel, but how did he feel about his final scene with Rorschach?
Did he regret killing Rorschach because he cared about him as a friend?
It's totally wrong to characterise Jon/Manhattan as being without emotion. Although we see him increasingly attempting to divorce himself from his own humanity (presumably to protect himself from the emotional impact of Laurie's affair) the reality is that he spends a considerable amount of time, effort and energy on creating a facade of implacability.
When it finally comes time to dispatch Rorschach, Jon hesitates. Even though he's seen the future and knows what must happen, he still attempts to reason with Rorschach. He practically pleads with him to be reasonable, to no avail.
The act is so upsetting that he lies about it a few minutes later. He can't bring himself to admit to Adrian (the mass murderer) the enormity of what he just did, that he just killed his friend:
And ultimately we learn (in Before Watchmen : Dr Manhattan #4) that Rorschach's death is one of the acts that has haunted him so badly that it, along with his complicity with Adrian's actions, convinces him to leave Earth entirely. So yes, I'd say that he regretted not just the necessity of killing Rorschach, but the act itself.
Recall Manhattan's words to Adrian/Ozymandias:
I've walked across the surface of the sun, seen events so tiny and so fast that they hardly can be said to have occurred at all. But you, Adrian... are just a man. And the world's smartest man poses no more threat to me than does its smartest termite.
It is important to remember just how far Manhattan is detached from humanity. If he regrets killing Rorschach, it is likely to be in the same way you or I might regret having to kill an insect.
It's also significant that Manhattan only got to know Rorschach after his transformation into a super-being. He maintains a somewhat greater emotional connection with those he knew as an ordinary human. The exception was Laurie, with whom he fell in love. But Manhattan was not nearly so close to Rorschach, whom he knew as more of an ally and colleague.
Conversely, Rorschach is very difficult for anyone to get to know, and when he interacts with Manhattan (for instance, when he sneaks into the military base at the beginning of the story) there is no indication the two of them feel any particular affection for each other.
Manhattan does hesitate for a few moments before killing Rorschach; but in general, Manhattan is reluctant to kill anyone. He might simply have been taking time to evaluate whether it was necessary to kill Rorschach to keep him quiet. On the other hand, he might have felt some human connection with Rorschach, perhaps for old times' sake, or out of respect for his courage and principles. As the readers, we really don't know.
In short: No, there is no reason to suppose Manhattan felt any particular regret.
This is distinct from what the reader is likely to feel; having got to know and understand Rorschach as a tragic anti-hero, we understand the necessity of his death, but may well regret it. Manhattan's perspective on Rorschach is quite different.