Does staring at a bright LED light damage your eyes? According to this article it seems that it is the UV part of the spectrum from the Sun that causes damage to the eye.
Would it therefore be "safe" to observe directly an equivalent energy density LED lamp, emitting in the visible part of the spectrum? 
 A: One can give a highly qualified, but definite "yes" in answer to your question.
Contrary to popular belief, if it weren't for the UV, then staring at the Sun would not be a particularly hazardous thing to do for the majority of people. This is why I said "highly qualified" - for people with certain conditions, simple staring at the Sun may be hazardous, even aside from the UV (more below).
There are two ways that light will damage your eye:


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*Thermal damage;

*Photochemical damage;


Fairly obviously from the nomenclature, the first kind (1) of damage is where so much energy is concentrated on the retina, its temperature is raised and the tissue is damaged or destroyed. The second kind (2) is where the light's photons are energetic enough to beget chemical changes by breaking bonds in organic molecules. This can lead both to acute poisioning of and damage / destruction to the tissue by the weird molecules / free radicals that come out of such light-matter interactions and also long term damage, even nuclear (in the sense of cell nucleus) changes, dysplasia (e.g. cataracts) and ultimately neoplasia (cancer).
The retina in the mammalian eye is superbly, densely envasculated. You won't find a better liquid cooling system in our contemporary human primitive technology. This situation has arisen from two evolutionary drivers: (1) the retina is simply adapted brain tissue, and the brain itself needs sophisticated liquid cooling: of all the organs it is the most sensitive to deviations from the warm blooded homeostatic temperature ($37{\rm ^oC}$) and (2) since we are creatures of the Neogene Eastern Africa, accidental looking at the Sun was everyday and commonplace to us. 
Therefore, if you stare straight at the Sun at high noon, your pupils will have shrunken to about $1{\rm mm}$ diameter, thus the power incident on the retina is of the order of $\frac{\pi}{4}\times 10^{-3}{\rm m^2} \times 1000{\rm W} \approx 1{\rm mW}$. This is WAY less than the normal, healthy retina's capacity to dump heat, even if concetrated into a diffraction limited spot. The uncomfortable feeling you get from looking at the Sun is mostly a psychological one: if you did it for thirty seconds, your retinal cone and rod cells would be so drained of ATP (energy stores) that you would be totally (albeit altogether temporarily) blind for many, many minutes. For either a predator or prey creature (we are both), this is not good. But the intensity alone, although it causes severe, but temporary blindness, is not a hazard (as long as you don't get eaten by a Neogene lion wearing sunglasses, or fall over a cliff texting on your phone whilst your sight recovers when the receptors finally get their ATP levels topped up). As you rightly point out the danger from the Sun is UV: it is not the intensity with the uncomfortable, squinty-eye feeling that arises from it that is the danger for someone with a healthy retina. It is the low level but constant UV dose one gets mostly from scattered light that is the problem. This is why sunglasses in many countries must by law fulfil stringent UV attenuation performance standards: sunglasses stop the "glare" but this is not the danger; indeed the "comfort" afforded by non-UV attenuating lenses is a very false sense of security.
Three other points about the Sun's power delivered to someone who stares at it. 


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*You have likely heard that looking at a total eclipse can be dangerous. It indeed can be. And this is because the pupil responds to average light. During a total eclipse, the pupil swells to its full size owing to the twilight's low average intensity. Its diameter swells to about $7{\rm mm}$: so the aperture has fifty times the area it has just before the eclipse begins. If you look at the diamond ring just after totality, you can therefore cop a dose of $20{\rm mW}$ or so in the eye. This can be enough to cause thermal damage. Evolution didn't kit us out to look at total eclipses;

*Some eye conditions mean that even the thermal loading on the eye from the Sun can be dangerous. Macular degeneration is a major one, as is albinism or even an extremely white complexion. Heart and circulatory diseases are others. Other diseases and defects mean that the pupil cannot respond to high light levels. Many recreational drugs can dilate the pupils severely, particularly hallucinogens like LSD, psylosin or mescaline can lead to eye damage in this way.

*Interestingly, if you begin with the assumption that the Sun's intensity into a fully shrunken pupil of someone staring straight at the Sun represents a safe upper limit to power dose (in the absence of UV), then you come up with dose limits that pretty closely match the ISO60825 laser safety standards for visible light. 
The last comment is particularly relevant to you. ISO60825 does NOT tell the difference between coherent and incoherent light. You tread an LED exactly as you would a laser: if you apply ISO60825 and determine that the light dose from your LED is intrinsically safe (i.e. class 1), then this is a sound indication, aside from in severe cases of the diseases I mention above. The other factor I haven't mentioned is the blink response, which is also accounted for by ISO80625, but this sets levels for class 2 and class 3A, which are not intrinsically safe, but which are deemed to be low enough that an accidental looking into the beam will not be harmful owing to the shielding afforded by a healthy blink response (assumed to limit the light dose to 0.25 seconds). Again, drugs severely interfere with this reflex.
A: In a nutshell, if it's bright enough, any light can damage your eyes. It is the intensity of light that is damaging (at least in most cases). 
