As far as I understand, when we view the ISS, it is because the sunrays get reflected from the solar panels of the ISS and reach us at the appropriate angle. So that would be equivalent to viewing the Sun through a mirror except that solar panels are not 100% reflecting mirrors.
So, is it safe to watch the ISS with the naked eye on a regular basis?
It's safe.
The ISS is about 100 meters across at its widest, and it's 400km away; this calculator tells me that makes an angular size of ~0.014 degrees. The sun's angular size is about 0.53 degrees. If both the ISS and the sun were square, the area ratio would be some 1400:1; the aspect ratio of the ISS is in roughly the ballpark of π/4, so the non-squarenesses pretty much cancel out, and you'd expect a perfectly reflecting ISS to hit your eye with something like 1/1400 the brightness of the sun. As noted in other answers here, the ISS is nowhere near perfectly reflecting.
Maybe the reflected light from the ISS could still be damaging if it was focused on the same part of your retina for a long duration, but there are two things protecting you: first, the ISS travels across the sky at a pretty good clip, and second, even if you try and track ISS with your eyeball, your eye's involuntary saccades keep the image of the ISS moving around your retina rapidly, spreading the received light energy.
Not only is it perfectly safe with the naked eye, it's also not a problem at all with a large aperture telescope.
At first, I tried looking at the ISS with a 80-200mm f/2.8 Nikkor attached to a scope converter. Handheld, it was pretty hard to find the ISS at all, let alone track it for more than 0.1s.
I then bought a small dobsonian telescope (Orion XT4.5). With a bit of patience and practice, I could watch the ISS at 135x magnification for a few seconds. It helps to have an assistant looking through the finderscope. If the ISS goes too far away from the view through the eyepiece, it can be really hard to find it again alone. The assistant can move the telescope back on track and you can then finetune the movement while looking through the eyepiece. The dobsonian mount makes it relatively easy to track the ISS while still maintaining a stable view in the eyepiece. It's tricky at first but very rewarding!
With a clear sky and ISS passing close to our location, I could perfectly see its H shape. The ISS in itself was never bright enough to be dangerous for the eyes. It can cause glare with a low magnification though (e.g. 20mm eyepiece for a 45x magnification). It's obviously much easier to track the ISS but it requires either a moon filter or sunglasses to better see the details.
It helps a lot to have a good eyepiece with a large apparent field of view. I used an Explore Scientific 82° 6.7mm. This way, the ISS stays much longer in the field of view and it's easier to track.
Last year, I bought a larger telescope : a 10-inch Zhumell Z10 dobsonian. Looking at a full moon with this aperture is painfully bright. The telescope is brighter, it supports higher magnifications, it shows more detail and has a smoother rotation than the smaller one. Still, the ISS was bright but not too bright with it.
On a really good night last July, I could count the 16 photovoltaic arrays and see the whole middle structure with distinct modules. That really was an incredible sight!
In addition to Russel's answer:
The ISS' albedo is far lower than 1, so the brightness is going to be less than 1/1400.
For the ISS to be dangerous to look at, it'd have to be as bright as the sun, meaning it'd be visible during the day instead of just another star at night.
The same goes for other objects too. The Moon is far larger than the ISS, but it's safe to look at because it's not a perfect mirror.
Here's a closeup of one of the arrays:
On the lower section you can see there's a zigzag pattern. The upper should be zigzagging as well but was damaged during a deployment.
Not visible in this image: I also get the impression the end of the array is not always perfectly aligned with the base of the array: the telescoping truss allows a certain amount of rotation between the ends.
There's a lot of poor information out there regarding looking directly at the sun (or looking at it through a mirror). As a general rule-of-thumb, it's a bad idea to look directly at the sun, but that is very overstated and the effects that it has on the eye depend on some very clearly-defined factors.
It's not actually harmful to look at the sun briefly - otherwise every human (as well as all apes and probably many other mammals) would accidentally blind themselves during early childhood. But there are several different types of eye damage that can occur.
1: over-saturation of the receptors. This is the most common type of damage, and it is the bluish or greenish thing that appears in your vision whenever you look at the sun or any bright light for a few seconds, or after a camera flash. The small light receptor cells in the retina can only output a certain amount of signal when they receive light. If they receive too much or receive it for too long, they get a bit overloaded and need to rest (they're cells, they can only handle so much). You will tend to see a reversed 'afterimage' (like a photographic negative) while these cells refresh. This is generally regarded as harmless, though it's probably not a good idea to overload the cells frequently because that's simply not how they are supposed to be used. Some more information is in the first section at http://www.healthguidance.org/entry/9913/1/Spots-Dots-and-Floaters-Seeing-Whats-Inside-Your-Eyes.html
2: burning or extreme over-saturation of small portions of the eye. This is the primary concern with lasers. A bright small light can cause direct damage (not mere overuse, but outright damage) to small parts of the retina, either through radiation damage or by concentrating heat on a very small area. This WILL cause long-lasting damage; sometimes it can heal over weeks or months, but sometimes it can be permanent. However, it is only to a portion of the retina, so the eye will still work, but vision won't be as good, and it can create significant blindspots. A powerful laser can do this, and looking at the sun through binoculars or a telescope can also do this on a rather large portion of the retina.
3: burning of the entire retina - this is the type of damage that people worry about the most from the sun. If you stare at the sun with your naked eye for a long time (several minutes to several hours), you can overheat the entire retina and kill many of the cells just from the heat. As the eye is mostly water, and has blood vessels that circulate liquid, it is good at handling and dissipating heat. But it can only handle so much. Additional cells can also be killed by the UV radiation that the eye concentrates onto the retina - so those cells may die hours or days later in addition to the ones that were killed more quickly by the heat. Most of us have the instinctive reaction of not staring directly at the sun, so this isn't a problem, unless someone forces themselves to do it.
4: long-term UV damage. If you stare at the sun often, or are in a profession where you are often at very high altitudes (such as a pilot) where UV isn't filtered from sunlight as much, or often use a tanning bed without eye protection, then you can accumulate damage from gradual UV exposure. The UV damages the cells on the molecular level. This damage is very gradual and often isn't instantly noticeable, and sometimes isn't even felt until well after the event because the damage can be in forms that prevent cells from repairing or reproducing. This is a problem common among old career pilots from the era when cockpit windows didn't have anti-UV tinting. It also tends to tie into other vision-reducing problems, because it's radiation damage and basically radiation damage harms everything - hence it will make any other (or latent) problems worse.
For the sun, the real worry is from the last two cases. For your question, the panels don't focus light onto a tiny pinpoint, so you won't get any damage from the concentrated light. And the light travels through the atmosphere on its way down to you, so most of that UV radiation is filtered like normal, in addition to its the dispersion in the reflection. And the panels don't reflect nearly enough light to overheat any part of the retina.
So you CAN look at the sun, as long as you follow certain precautions: don't stare at it for a very long time (and especially don't hold it in the same part of your vision if you do), don't magnify the amount of light that your eye collects (like with a telescope or binoculars), don't concentrate that light down to a very small point, and don't look at it from very high in the atmosphere.
Source: I spent many days trying to get real answers about eye damage before the eclipse, because "don't look at the sun. Ever." is simply an illogical statement when you consider that all land creatures occasionally glance at the sun accidentally. I can dig up specific, more credible sources if needed - but many of those sources will be generalized statements from eye doctors (at least for the first 3 kinds of damage) - medical studies almost always focus on very, very specific things, whereas these types of eye damage are much broader and beyond the scope of any single study. The closest study to "does looking at the sun for a long time cause eye damage?" was a report accumulating eye damage data from the prior eclipse, where patients went to the doctor after they got vision damage because they stared at it for too long (thus, a very self-selected group, and no one knows how many people watched it without eye damage or without going to the doctor)
First, the assumption that we can see ISS because its solar panels reflect the sun, like a mirror does, is incorrect. Some satellites, most notably Iridium, are known for specular reflections, called flares, but I haven't seen any reports of ISS doing this. In the best case it is an extremely rare event and I haven't never observed it, despite seeing ISS occasionally.
So, now that we know that ISS is more like a regular everyday object illuminated by the sun rather than a mirror, the answer is obvious:
Can we safely see sunlit objects with the naked eye? Yes, we can.
(Yes, it would be safe with a mirrory sun reflecting ISS, too, but that's a different question!)
