I don't think it will heal, but, if you wanted to leave them the way it is, I don't think you're going to hurt the plant. I've heard that, if you want to get rid of those leaves, to cut a nick into them. This will cause the plant to pull the moisture back into the plant, and withering the leaf. The withered leaf can then be removed without damaging the plant. I've never tried this though. If you decide to cut the leaf off, I suggest sprinkling some cinnamon on the cut afterwards, it acts as an anti-fungal agent.

There seems to be some confusion again with common names - Aloe vera is Aloe vera :-) Although that term is sometimes used to refer to ANY kind of aloe, there are not two species of true or common, aka medicinal aloe. There ARE many species of aloe....as many as 400 have been registered but true aloe, Aloe vera (formerly Aloe barbadensis), is the one noted for medicinal purposes. There is some documentation that Aloe arborescens also has similar medicinal properties but as the species name suggests, this has a much more tree-like growth habit and develops a distinct stem or trunk. The difference in growth habits between the two plants shown in the pics is simply due to age and maturity. Aloe vera will form a full rosette if given sufficient time and proper growing conditions. Check the Wiki photo linked below. The smaller ones that appear to only grow laterally rather than in a round rosette of leaves are just young, immature plants (offsets or pups) and hampered from full development by lack of time and indoor growing conditions. Here is a link that might be useful:

Karen it was stated that it is an Aloe Vera, they grow in Africa in hot dry climates, storing water in their leaves. They wait for the yearly rainfall to store water again. If they live in a fog belt they will benefit from the CAM theory. taking in moisture at night. With New Yorks humidity it should be kept very dry and grown in a clay pot with fast draining coarse soil. NO PEAT MOSS. Karen sorry I don't agree with you. They still come from Africa and not from Wal Mart. Came originally from No. Africa and Arabia, but now are bing grown in other countries that have warm winters. Must not have Temps below 32% or live under protection of other trees and shrubs. Go with the directions of more light. They are accurate. If the drooping leaves bother you cut them off as soon as it warms up where you live. Not now, those dead leaves will protect the plant from the cold. Now if you lived here, I would trim all the leaves off that were going down. Get a new pot with proper soil, cut off the head of the plant let dry and just set on top of the soil to reroot. During the begining of the hot summer. Norma

L, Aesthetics aside, they're not unhealthy as long as the plant's healthy. They eventually fall off. ETA - your plant looks in the pink of health. As stated, the overweight leaves are a case of too much water/not enough sun, air movement and higher temps. Aloe vera is thought to have come from the Horn of Africa/Arabian Peninsula area (its origin is unknown), and grows quite well under harsh conditions. My recommendation is to back off on the watering to 2-3x a month - that would allow those leaves to lose their overabundance of water.