If you wanted to give evidence that Eveline is not in love in Joyce's story named after her, you could cite her deciding not to leave Dublin and go to Argentina. She opts to stay with her father and brother and remain in Dublin, instead of starting a new life in Argentina.
I wouldn't really recommend making that argument, though. First, it's beside the point. That's not what the story is about. Eveline is another paralyzed Dubliner in Joyce's short story collection, Dubliners, who can't break away from her mundane, isolated existence, even when she has the opportunity to do so. She is stuck in a role as caregiver, with no future of her own. She has a chance for self-improvement, but freezes at the critical moment. She is a Dubliner trying but unable to break out.
Second, the idea that she stays because she doesn't love the man who wants to take her away is a misinterpretation. She stays because she is paralyzed, not because she doesn't want to go. She has a chance to break out of her trap, but fails to do so.
The story reveals little about the man or Eveline's relationship with him. This is significant. The relationship is not the focus of the story. Eveline's paralysis is.