One of the most divisive things in the BLEACH fandom is something that continues to be a point of contention and debate, even though the undeniable truth is evident in the series from the word go. Ichigo and Rukia's platonic relationship is seen as a cop-out by fans who, for some reason, have decided that the S.S. "IchiRuki" is the series' real ship over the canonical outcome which sees Ichigo eventually marry and even have a child with Orihime after the events of the Thousand-Year Blood War arc. Rukia, on the other hand, goes the friendship-to-romance route with her childhood friend, Renji Abarai, with whom she also has a child.
Ichigo and Rukia's relationship is definitely integral to the BLEACH storyline, as well as to the development of both characters; however, the reason for this is far more complex than the premise of some kind of romantic outcome for the two of them. Here's the run-down on the fated friendship of Ichigo Kurosaki and Rukia Kuchiki.
The Rain
A recurring title in the BLEACH narrative for various chapters, the concept of "Death and the Strawberry" as the introduction to BLEACH is important as it centres the connection between Ichigo and Rukia, alluding to the "fatefulness" of the encounter, as well as its significance. The first time "Death and the Strawberry" is invoked is the very first chapter and episode of the series, in which the 15-year-old Ichigo Kurosaki has his first encounter with a Shinigami, despite having long been able to see ghosts. The encounter ends up having life-changing effects as it becomes Ichigo's entry point to Soul Society, which in turn becomes a journey of self-discovery as he learns more about the world around him. For Ichigo, meeting Rukia is an important moment in time because she was able to "make the rain stop". In BLEACH, rain is a common device used to communicate Ichigo's feelings in moments of turmoil.
His inner world, which consists of countless sideways skyscrapers, is constantly besieged with rain whenever Ichigo is going through a tough time, something that even the spirit of his Zanpakutō, Zangetsu, has taken note of. The rain is especially furious in moments where Ichigo finds himself unable to protect the people closest to him, with the worst rain coming after the death of his mother. His most important character trait is his desire to protect those around him, and when he can't fulfill this central dogma, it completely ruins Ichigo's sense of self. It is this trait that makes Ichigo truly grateful to Rukia for saving his life, and enabling him to defeat the Hollow and protect them. After the battle against Aizen, Ichigo's reiatsu begins to dissipate as part of the payment for his use of The Final Getsuga Tenshō. Ichigo loses the ability to perceive Rukia and reverts to a normal human; and during the events of the Fullbring Arc, Ichigo was manipulated by former Substitute Shinigami, Ginjō Kūgo, who preyed upon his desire to return to his former self.
Guilt and the Strawberry
In saving his life and the lives of his family members, Rukia earned Ichigo's respect, a currency which went on to be invaluable. Despite nearly losing his life in the fight against Renji and Byakuya, Ichigo risks his life again to go to Soul Society and rescue her – because that's what he felt he owed the person who saved his life. While Rukia's influence on Ichigo is clear, his influence on her is perhaps the most important reason why they are so close. Rukia's central dogma is revealed during her incarceration in Soul Society – her biggest regret is the death of the former Lieutenant of the 13th Squad, Kaien Shiba. While Rukia's hand is the one that slew Kaien, his death was actually the result of Aizen's Hollowfication experiments, which are in turn immensely significant to the development of Ichigo and Rukia's relationship because it is Aizen who gets Rukia sent to Karakura Town in the first place.
Despite there being no other way to deal with Kaien's Hollowfication, Rukia's trauma and guilt at not being strong enough to save one of the few people who saw her for who she was and not just a member of one of the Great Noble Families of Soul Society. Rukia's grief would lead her to bring Kaien's corpse to his family, but because she blamed herself for his death, she would also go on to tell the remaining Shiba Clan members that she is Kaien's killer. In the earlier days of Soul Society, there were five Great Noble families, of which the Shiba Clan was one of them until its fall from grace following the death of its most prominent living member: Kaien Shiba. Effectively, Rukia saw herself responsible not only for the death of her friend and superior, but also for the subsequent collapse of the Shiba Clan, which she didn't know was actually a two-pronged incident beginning with the exile of Isshin Kurosaki née Shiba. When she was arrested and put on death row for giving her Shinigami powers to a human, Rukia battled deeply with the feeling that this outcome – her death – is ultimately what she deserves. This guilt that she carries while imprisoned; however, is augmented by the fact that deep down, Rukia didn't really want to die – which only made her feel worse.
Seeing Double
Retrospectively, it doesn't make much sense for a Shinigami with Rukia's competence to be wounded while protecting a human from a Hollow attack. Rukia could have beaten that first Hollow easily; however, she had to deal with protecting the reckless Ichigo from getting himself killed, and in trying to explain the situation to him mid-combat, Rukia was greatly distracted. However, she had no real reason to keep him from getting killed by the Hollow – Shinigami have been known to ignore certain distress calls, as was seen with Kurotsuchi's involvement in the delayed reaction to the Hollow attack on Sōken Ishida – Uryū's grandfather. If Ichigo had been used as bait, Rukia could have even killed the Hollow with no casualties, but something compelled Rukia to not only try to protect the teenager who ignored all her warnings. For one, Rukia felt guilty again for involving Ichigo in the battle, but also because she accidentally galvanized him when she told him that he was the Hollow's real target.
In the beginning, Rukia doesn't appear to notice that Ichigo looks exactly like the subject of her greatest guilt, but this could be attributed to her dulled state at the time. In the Soul Society arc, Captain Ukitake sees Ichigo for the first time and is so shocked he can barely speak. Ichigo's resemblance to Kaien is uncanny, and close enough to earn shocked looks, and even some level of confirmation from Byakuya himself. It doesn't make sense that Rukia would meet Ichigo for the first time and not be just as shocked as Ukitake was. It would then follow that the big motivator that spurs Rukia into action; that made her want to protect Ichigo at all costs; that even led her to offer him her powers despite it being a huge Soul Society taboo, is the fact that this orange-haired boy resembles Kaien.
Absolution
In saving Ichigo and the Kurosakis, Rukia is effectively doing what she couldn't in the past, and making up for her guilt, even if she doesn't realize it yet. Ultimately, Rukia's actions that day set into motion a sequence of events that would have her and Ichigo rely on each other deeply, and foster the kind of friendship that transcends the guilt.
In making the rain stop for Ichigo, Rukia puts an end to her own suffering as well, but at no point is this because of a romantic connection between them. In the end, Ichigo going with Orihime, who is in many ways like his mother; and Rukia going with someone with whom she has seen both life and death, just makes sense.