Iapitus The Impaler said:
He can time travel in time stop
And Heaven Ascent dio can time travel as well. However that doesn't matter because Heaven ascent dio is a reality warper, which means as long as he wishes to cause something it will happen when he uses reality over write. And as he showed in the battle against GER, he has no qualms going head to head with other abilities and abusing Reality overwrite in character (As in a review Araki confirmed, GER would have won that fight if Dio didn't immediately use his reality overwrite to negate GER's effect). Also minor nitpick, the Stalker cannot timetravel in timestop. He can simply choose to switch from stopping time to reversing or fastforwarding time, meaning he is not stopping time as he time travels and is vulnerable in these time periods.
Now then, to go through every little issue I have with this thread (As I warned you I would do this) thanks to Reality overwrite. Start with auto-restoration through rewind. Immediately negated by Reality overwrite, as Dio in canon has shown he immediately removes any possibility of rewinding time or themselves in any way shape or form on the first hit. (Defeating GER feat without prior experience anyone?) When he recognizes a threat, killing people straight up via reality overwrite is something natural for him to immediately do. So getting past Stalker's time rewind according to the feats is something that shouldn't have to be discussed.
Now then, second issue is arguing if Dio could see Stalker in spirit form for possession. Answer for this in canon should in fact be a yes thanks to Arc 4, Diamond is unbreakable. For those who don't immediately understand or never read/watched Diamond is unbreakable, the context is that there was a ghost character called Reimi who spends more than a decade searching for someone she could interact with. Reimi is in the same state of being as the Stalker is when he goes into spiritual form. However, while no normal humans could interact or even see Reimi, Stand users had no issues seeing her and even interacting with her using their stands. (although she could still go through them all the same)
Biggest example is Heaven's door affecting her, which showed that she wasn't a stand user but could be affected by a stand just like any other living being. As far as its been shown, only stand users are able to see and affect spirits/ghosts in canon thanks to their stands being of the same type of being as ghosts (To tone it down for those who don't understand the concept. You know how only a stand user can see other people's stand? The logic of Araki is that only stand users can see people's spirits or ghosts, which aren't all that different in mechanics as stands honestly). While if it had just been the event above I wouldn't believe all stand users had this ability, at the end of Part 4, every stand user sends a goodbye to Reimi before Reimi leaves to move on. Every stand user from Morioh, meaning it must be something every stand user can see.
As for the kc thing, that whole part has issues. While kc moves when there is time erased, you have to keep in mind everyone moves in this time period. Its just that no one but the kc user is able to break out of fate's chains for what they are scripted to do before the time erase space occurs. So if moving in this space gave you infinite speed, everyone who survived to any point of a fight with KC, Polnaref, Bruno, Mista, etc would have infinite speed. The only feat you can gain from this is time manip resistance if you are able to consciously see and move within the time erase portion honestly. (GER doesn't just resist time erasure mind you, he resists everything short of reality warping from what we know of his appearances in manga + games thus far. Even Araki points out if Dio hadn't in char overwrote reality against GER, Giorno could of stomped him just as hard as he could stomp Pucci and Diavolo).
As I don't see anything the Stalker can do unless he starts gaining feats of travelling across universes and pulls a D4C on Dio like Jotaro did, and then still managing to kill him after that before the hands regrow, the Stalker is stomped out of reality by Dio.
(I had more to point out at the beginning, but one wifi not posting post later and losing mose of what I wrote I ended up with this instead) Note: Even if Dio doesn't erase as an opener in char, he still reality overwrites as an opener in char.