Why do these nectarines taste like aspirin?

Salicylic acid is found in all fruits, vegetables, and herbs to some degree. Salicylic acid is close in structure to acetylsalicylic acid, also known as aspirin. Perhaps they taste similar. However, the content of salicylic acid in the food is typically very low. The study from Wiley Interscience found only 0.02 mg/kg in kiwifruit at the low end to 0.10 mg/kg in New Zealand grapefruit at the highest. To put this in perspective the recommended adult dose of aspirin is 275-300 mg.

Perhaps you have some weird nectarines where the concentration of salicylic acid was high enough where you could taste it. Salicylic acid is a plant hormone that has a variety of purposes including defense against pathogens. So perhaps your tree was experiencing an attack by some invader and thus salicylic acid levels were spiking.