Daisy never sincerely tries to get Gatsby, so she can't truly "give up" on him.
The Great Gatsby is a story of, among other things, the empty dreams and empty lives of its characters. Daisy is a prime example. She first meets Jay Gatsby when she is an eighteen-year old who enjoyed entertaining all the young officers stationed near her parent's home. Her parents prevented her from going to New York to "say good-by to a soldier who was going overseas" - it's implied this was Gatsby. The relationship did not survive while Gatsby is in the war, although she nursed melodramatic and romantic illusions about what might have been.
Marriage to Tom proved to not be satisfactory, either - he was abusive and unfaithful. When Gatsby and Daisy learned of the physical proximity of their residences and the social circles they shared - Tom and Daisy because of their wealth, Gatsby because of the charade he maintained about himself - their mutual attraction renewed itself. However, Daisy loved Tom at one time and loves his money at all times.
Gatsby walked over and stood beside her..."Just tell him the truth-that you never loved him-and it's all wiped out forever."..."Oh, you want too much!" she cried to Gatsby. "I love you now-isn't that enough? I can't help what's past." She began to sob helplessly. "I did love him once-but I loved you too."
In the end, Daisy went with Tom as they moved away from the area, the associations, and the memories.
They were careless people, Tom and Daisy-they smashed up things and creatures and then retreated back into their money or their vast carelessness, or whatever it was that kept them together, and let other people clean up the mess they had made.