Leaving fruit on a plant until it is fully ripe is more than just a vague command by garden centers; letting the fruit become completely ripe means it is at the height of taste and sweetness. Most fruit contains sugars that make the fruit tasty. These increase as time goes by, but that also means the overall sugar load you get when you eat the fruit increases. However, that does not mean you have to avoid fruit completely if you are trying to avoid adding sugar to your diet.
It Does Increase
Sugar content increases in most fruits as the fruit ripens. An increase in sugar and sweetness is part of the ripening process when the fruit is still on the tree or vine; it is not something that happens separately. In fact, sugar may not only stop increasing once the fruit ripens, but it can actually decline instead. Some fruits continue to ripen after being picked, but in those cases, the sugars do not increase.
As far as the fruit and plant are concerned, sugar content in fruit increases for the sole purpose of attracting creatures that could spread the seeds around. Birds, rodents and even sloppy humans can all leave the seeds that were inside the fruit along the ground, helping the plant reproduce. If the fruit were to remain too sour for creatures to eat, the plants would have a harder time spreading. Even lemons have their predators; squirrels will occasionally go for those fruits, even though the lemons are not always a squirrel's favorite.
If you pick fruit too early, especially fruit that doesn’t continue to ripen after it’s been picked, the fruit won’t taste as good and could even be unhealthy. Purdue University says pineapple tends to be fairly starchy until just before it becomes fully ripe, and unripe papaya has a high latex content that can be irritating. While unripe papayas are sometimes part of Southeast Asian cooking, they are cooked, and not eaten raw like ripe papayas can be. Fruit is meant to taste sweet, and picking it early only means you have unripe, undesirable fruit instead of low-sugar fruit.
Role in Diabetes
Diabetics are allowed to eat fruit. The Harvard School of Public Health does note that the higher the sugar content in the ripened fruit, the higher the glycemic index. However, that does not mean the fruit has a lot of digestible carbohydrates. Harvard uses watermelon as an example; the water-filled fruit has a high glycemic index but a low amount of carbohydrates. Those with diabetes should speak with their doctors about how to calculate glycemic load, a measurement that looks at the combination of glycemic index and carbohydrate amount, as this will help them fit foods like fruit into their diets. Simply picking unripe fruit is not going to help someone fit more fruit into their diets.