This is something I've wondered for a very long time. So, Obi-Wan Kenobi manifests himself as a Force ghost fairly often. We all know that Luke can see him and later other Force ghosts. But can R2-D2 see Obi-Wan in ROTJ? I mean, can R2 even compute what he is? I assume he can see him, since he's looking in that direction as Luke walks toward Obi-Wan. But does R2 comprehend that he's an apparition? There's something odd about droids being able to see what is essentially a phantom. R2 does have a great deal of personality, given his lack of memory erasures or repgramming. We see this throughout the EU and even as late as Star Wars: Legacy. Definitely seems to understand the Force, the Jedi, and Sith more than C-3PO. He even refused to show Luke and his family his recording of Anakin/Vader Force choking Padme, and I can't say I blame him.
I don't think non-Force-sensitives, or untrained Force-sensitives, can see the ghosts, let alone any droids.
Droids are purely mechanical, certainly not as advanced and tuned into human emotions as replicants. In SW they are basically that.... machines. They are tools, designed and programmed for a set of specific functions, with AI programmed into them to interact better with organic beings. Any sort or mysticism or religion, which I consider the Force to be part of, would be way beyond their programming. In fact, I think even Alien androids like Ash or Bishop are more advanced and much more sensitive to human emotions, meaning greater effort was put into programming them that way, than SW ones. Where in SW do we ever see a real humanoid, meaning one that could pass for an actual organic creature? In SW they are tools or slaves/soldiers/pilots..... whatever, not supposed to interfere with or have real emotions. I'm talking strictly Lucas canon here, so don't try to get me started on EU stories. But if we are getting into "real" Earth ghosts here, this whole discussion will go way off topic. Supposedly some (organic!) people claim to see/feel them while others don't. To my knowledge, "photos" have all proven to be fake. So the average SW droid.... I'd say nope.
Artoo has shown himself to be a very unusual droid. I think he sees and knows much more than we give him credit for.
It's an interesting debate. He certainly understands human emotion and interpersonal dynamics far more than 3PO. He seemed genuinely saddened that Anakin stopped talking to him once the latter became Vader, and I would theorize that he recorded Anakin's assault of Padme in case she needed evidence or something. I presume he was very freaked out. He was very good at keeping secrets (a la knowing Obi-Wan, Luke and Leia's siblinghood, parentage, etc.), which leads me to believe that he was disturbed by Leia's kiss of Luke in TESB. Anyhow, as to whether or not R2 can see Force ghosts like Obi-Wan, I tend to lean toward the notion that he can. But it's something to contemplate.
Can any droid? No. Can Artoo? Yes. Artoo can also sense unease coming from the cave on Dagobah. Artoo is the goat; just accept this.
Artoo is what gives the Jedi his power, he's a robotic energy field, created by all metallic things, he rolls around us and penetrates us, he binds the galaxy together'
It could be that doids don't see them, but do get some kind of weird reading on their sensors, like the shows where people try to find ghost activity on their devices.
Poe Daemron: “Somehow, R2 can see dead people.” R2: *beeps* Poe Dameron: “And you never told Yoda this why?” R2: *ignites red lightsaber* “You know too much…”
Even Leia didn't seem to see the Force ghosts at the end of ROTJ, so I highly doubt any non-Force sensitive being, let alone a droid, would be able to.
Artoo cannot see a ghost, but like Threepio can see an illusion. His reaction to Luke going into the cave stems from listening to conversation about what it is and understanding the seriousness regarding it. He didn't follow Yoda when he went off towards there years ago, so he's not clear on what happens.
Wait, when did Threepio see Luke go into a cave? If we’re talking about the trick Luke pulled in The Last Jedi, remember everyone saw him and thought he was actually there until it became abundantly clear he wasn’t after the barrage of Imperial fire didn’t reduce him to red paste.
R2 suffered severe head trauma in ANH. He basically got JFKed. Who know what he sees. There are plenty of apocryphal tales that tell of spooky abilities been bestowed upon the cranially injured. He probably does see ghosts, and a whole host of other equally terrifying stuff. He's like Tim Robbins in Jacob's Ladder.
As far as new canon is concerned, I don't know. And I still don't know why he didn't have much to do in the ST. In Legends, I believe Cade Skywalker owns him. It's been several years since I read those graphic novels, however.
Frankly, Luke's appearance and trick at the end of that movie is one of the very few things I actually like about that story. I would have preferred him to actually be present and perhaps deflect some laser blasts or something. And Kylo Ren is acting like an absolute madman, which goes with how melodramatic and absurd that movie is. But anyhow, it was impressive that Luke's illusion was powerful enough for 3PO to see him. As for weather or not R2 can see Obi-Wan as a Force ghost in ROTJ, I think it's definitely possible. He may not understand what he's seeing or may comprehend it. I don't know. But Leia doesn't appear to be able to see Anakin, Yoda, or Obi-Wan at the end of ROTJ. I assume this is due to a lack of training at this point, even though she did have latent Force abilities when one considers her resistance to mental probes in the Obi-Wan miniseries, her concealing information in her mind from Vader in ANH, and, of course, her being able to hear Luke in TESB. So, it's up for debate. However, I can see why R2 wouldn't be able to see Obi-Wan in ROTJ because Obi-Wan manifested himself to talk with Luke. R2 may have been wondering who Luke was talking to and so on, but he doesn't say anything, and that leads me to believe that he understood what was going on.