Throughout the entire series we see that Asuna is Kirito's world and vice versa, the whole reason many things in the universe happen is because of this, but the way I see it is Asuna is just there as someone who is just filling a void left in Kirito that can never be filled. The void that was left when Sachi died in front of his eyes and to this day he still feels he is the one solely responsible for her death. Anyone who has watched Sword Art Online knows that Sachi was only in the show for maybe 1 and a half episodes before her demise, so how could she have made that much of an impact on him ??
The way I see it is simple, once Kirito joined the Black Cats guild he found companionship and acceptance, while he was protective of everyone as they were a much lower level than himself he was mostly protective of Sachi and I believe a lot of that had to do with him developing feelings for her. Not fast and "in love" already but they were there starting to grow and I think that the conversation the two share underneath the bridge continued that growth or progressed it further. Her fear of death, the fact that she even joked about a double suicide, but his promise to protect her and help her get out of the game and return to normal. When she comes to him later, still sad and unable to sleep, his words: "If you stay in this guild, you'll be safe. You'll definitely be able to go back to the real world someday." and, "It's okay. You're definitely going to survive." show his unwavering need to protect her. It may seem like the guild but Sachi was the only vulnerable one and had opened her heart to Kirito. It's not easy to explain a first love, that I can tell you for sure. However, it's easy to see when it's in front of you though presented in a show, you can tell when that happens and you can tell with how much she becomes a part of his future that he truly does care about her.