Having lost to four different Rosalina players this year, I can safely say Rosalina vs. Ike is 6-4 in Rosalina's favor. The easy part is getting Luma away from Rosa. The hard part is finishing the job.
In the neutral, Rosalina's Fair and Nair have sufficiently large hitboxes. You cannot get in on her, even with your own disjoint, and her aerials have little to no endlag. Trying to wipe her out, on the ground or in the air, comes down solely to her own mistakes. Her stalling ability is among the best in the game, especially given the speed of her dashgrabs, dash attacks, and the knockback on her jabs.
Worse yet, her rolling is the safest I have seen from any opponent in the game. I feel her roll seems to end early, allowing her to shield, spot-dodge, or jab before Ike can chase her down for a dashgrab punish, resulting in him getting punished instead. Her safety and lack of endlag alone makes the difference in a matchup with someone who is not safe on his own attacks and has more startup.
When she hits with a UTilt or a Uair, there is no way to avoid the juggle. If you've already used your airhop to attempt an escape and you still get hit, Ike's options rest in Rosalina's mercy. It takes an extremely well-timed airdodge and not having it read to avoid it. If you're on the ledge, you can Aether to the ledge. If you're center stage, you can combo Ike to a sufficient percent. As I faced in a match with Gadiel Vastar last night, he can chain the Uair from 70% to death on Dream Land. This is due to a combination of low ceiling, platforms, and Ike's lack of sufficient aerial movement and options while in tumble.
While Ike is offstage, her edgeguarding options include Dair spike above Ike's Aether and Side B, and extending the range of Luma's FSmash to right past the edge. She can also knock a Bair while chasing a falling Ike.
While Rosalina is offstage, her options are not as limited as they seem. Despite her slow falling speed, it's advantageous to her as Ike is not the type of character to chase people off-stage for the most part. This can result in getting counter-gimped, or she can simply Up B past him, recover, and prep a Luma edgeguard as he recovers on stage. Her ability to snap the ledge quickly with Up B makes it hard to time an Eruption, while her ability to shark up the ledge with Uair makes it less of an option. Rosalina is also volatile on the ledge, as she can not only quickly attack while jumping, but the Luma can just do it for her while she jumps from the ledge, making edgeguarding a jump difficult. Her ledge attack also hits shorthops, so retreating Fairs are also hard to accomplish.
Ike's main tools in this matchup cover Rosalina's light weight vs. his heavy weight, his own disjoint while works while Luma is away, his kill power, and his edgeguard. Rosalina's tools include Luma killing power, Uair juggles, abilities to counter edgeguarding, dashgrab speed, dash attacks that perfectly punish rolling due to their active frame, and having clear advantage on a number of stages.
Battlefield, Dream Land, Town & City, Halberd, and Delfino Plaza are all in Rosalina's favor. BF, DL, T&C, and DP give her platforms to carry juggles, while DL, T&C, Halberd, and parts of Delfino have low ceilings. Halberd and Delfino allow her to safely "shark" free Uairs under the stage, while Delfino also allows her to have clear control in the neutral without the risk of being edgeguarded on certain parts of the stage. Final Destination, Castle Siege, and Smashville are the only stages that break even for Ike in the matchup, but if any of these are banned, the stage matchup once again favors Rosalina entirely.
Therefore, given stage advantage, Uair juggles, low ceilings, safety, and Ike's own lack of aerial and tumble options while being juggled, I firmly believe the matchup is 6-4 Rosalina, which is the same as Sheik's matchup ratio on Ike.