Can you suggest any organic methods to control leaf curl in nectarines?
With leaf curl, it does come down to prevention, which is essential. It can be controlled quite easily by applications of copper oxychloride or lime sulphur at any stage during dormancy. As these products are natural, they can be regarded as an organic approach to this problem. The most effective time to spray your nectarine is when the buds are swelling but before they have opened.
Note; once the fungus has entered the leaves, there is no way to control leaf curl. If your tree is already infected, you can prevent it next year by spraying as above. It is an ongoing process to repeat each winter to stop re-infection.
Normal pruning techniques should remove most of the infected or dead branches. Garden hygiene in this case is very important, so remove all dead leaves that have fallen to the ground during the growing season, as this help lessen the chance of re-infection.