Can this citrus tree be saved?
Thank you for your question.
Two different types of symptoms are noticable on the photos.
1. Fruit with spots. This is likely to be something like citrus (sweet orange) scab caused by fungus, Elsinoe sp. or bacterial canker on citrus. Birds are also known to do such damage and there are a few places on the fruit (upon closeup) suggest large pecked-out areas. This is one which may require a closer look for better diagnosis.
2. black stuff on leaves is sooty mold. This is a saprophytic type fungus that grows on residue(honey dew) left by insect. Often times, you might find aphids and/or scale type insects on the plant. The sooty mold can be washed off, but one should treat for the insect pest to reduce future problems.
For #1, should the fruit be picked and discarded, or will the inside taste fine? Is there an organic treatment available?
For #2, we'll look closer to see what insect might be the source.
Often with scab, the damage is superficial or localized to the rind and does not penetrate into the fruit. Fruit should be okay to eat. However, if you see damage extends all into the fruit that result in discoloration or mushy fruit - DO NOT EAT THAT!
I am not aware of an effective organic treatment for scab.