If you truly love someone is it that easy to walk away? If it is, then maybe you didn't love them after all? (Just playing Devil's advocate).
If I really loved someone, dunno it depends, if they are prone to moments of self doubt and wondered how anyone could love them, then no I would just see that as one of their little imperfections and I would reassure them that I did love them or possibly question whether I was trying hard enough.
If I knew I was putting my heart and soul into it and still being accused of not trying or 'not giving a ****' then I think that would mean I would start to love them less and possibly get to the point where I wanted to leave. I've walked away from a similar situation very recently in fact but it was more the fact that I felt they didn't love me anymore that made me do it. I maintain my judgement was correct, even though it was crippling to have to go through it. To put it simply, I came to the conclusion that I'd still hear "I love you too" in response to me saying it but no longer heard "I Love you" coming the opposite direction. Man, what a horrible, heart destroying break up. *shudders*.