First let's look at what your acquaintance actually does to the diet itself.
Suppose that his normal TDEE is 2500 kcal. That means, at his regular activity level given his job and physique, training not considered, he'd need about 2500 kcal per day to supply his body with the energy it needs. Eating that, there would be neither weight gain from excess energy being stored as fat and glycogen, nor weight loss from having to burn fat and use stored glycogen. Now suppose that on his diet he "consumes" a net 1750 kcal per day, either by caloric restriction, training, or a combination of the two. There is a 750 kcal deficit per day, which will have to come from somewhere since the body still needs its energy for functioning. Suppose he maintains this 6 days per week. That makes for a 4500 kcal deficit over those 6 days.
Now, if he then eats 7000 kcal on that remaining day, basically the whole deficit has been undone, since 7000 equals his regular 2500 TDEE plus the 4500 deficit built up over the week (see also note ). The exact impact will depend on how much is consumed on the cheat day, as well as the caloric deficit during the other days. If by diet you mean he just eats very strict and clean, but there's no deficit, then the cheat day is basically just a ticket to weight gain. The cheat day may end up achieving a caloric surplus over the entire week, an exact balance, or still leave a deficit. It depends on his basal metabolic rate, activity level, training and what is consumed over the entire week.
Now for your specific questions.
How does this impact your health on a physical level?
I can't imagine it is too healthy. Carbohydrate-rich food tends to lead to a spike in blood sugar (particularly simple carbs with a high glycemic index, like the sugar in ice cream). Because the spike is excessive, the pancreas has to secrete much higher amounts of insulin, which will bring blood sugar levels down by promoting glycogen storage in the liver and muscle tissue. Since the insulin spike is excessive too in response to an abnormally large carb load, once the blood sugar has been lowered it then dips below its normal level, leading to the so-called "sugar crash". Actually, ice cream might not be the worst thing for this. Because it also contains a lot of fat and some protein, being milk-based, it does provide some slower-digesting, useful macronutrients. That could offset the effects of the sugar somewhat. The blood sugar and subsequent insulin spikes are lower for foods with a lower glycemic index, often the case for complex carbohydrates such as whole-grain products, oats, brown rice instead of white etc.
Having the pancreas work so hard suddenly and having a large insulin spike isn't particularly good. Although I'm not sure if it is really problematic when this happens infrequently. The lifestyle problems that lead to type 2 diabetes are usually of a more pervasive nature. Also note that the body can only store so much glycogen. Unless the rest is used as energy (which could be the case if training is performed on this cheat day) it will be stored as body fat. Eating strict during the week and training while using the readily available glycogen storages, which will probably be refilled over the week as well (assuming this person isn't eating very low-carb outside of the cheat day) and then having a sedentary day eating a large surplus of calories when the only thing your body can do with it is increasing fat deposits doesn't seem like a great body composition strategy.
On a mental level?
Surprisingly, this might in fact be a good thing on a mental level. At least, if it's kept in check.
I don't really like the term "cheat day" if such a day is applied in a planned manner. "Refeed" is a lot better. On a refeed, during one or more days you eat at the TDEE level or above. This has physical and mental benefits.
First of all, it makes diet adherence easier. A lot of folks have this classical notion of a diet where you need to eat very clean, scarfing down stuff like salads and maintain a daily caloric deficit (often a big one). This is foolish. In my opinion, unless there is an eating disorder people will not win in a battle against hunger. Our drive for seeking food is one of the most basic, important instincts, based on biological factors that have been around a lot longer than our higher brain functions. Trying to use "willpower" against hunger is like pitting a karate white belt against an MMA fighter. You're going down. Maybe some folks will last weeks, and others even months, but you will go down eventually. A diet should be a manageable caloric deficit, eating good foods that fit a template which you'll wish to follow afterwards as well. So it's kind of a dialed-back version of what you'd normally eat to maintain weight, with some adjustments here and there. Where does a refeed come in? Well, even if you have a very manageable, realistic diet, caloric restriction and having to keep track of your food all the time is still exhausting, both physically and mentally. A refeed gives you a break. It's like a refreshing nap when you spend long days with not enough sleep.
Going back to the physical stuff for a moment (but this will also affect you mentally, it's two-way traffic) refeeds make sense too.
First of all, a prolonged caloric deficit is going to cause the body's metabolism to slow down, beyond what is expected from decreases in body mass. This is mostly a factor in severe caloric deficits, but even so it's still something to keep in mind (see note ). If there is a day every now and then where you indulge or eat over maintenance you are showing your body "See? We're not starving or in a food crisis. Keep going." It isn't unexpected that your body would downregulate metabolic function during a food shortage, just like how you can save on fuel in your car by driving a bit slower; fuel consumption doesn't rise completely linearly with speed (there's a point of diminishing returns) so if your tank is getting empty, sacrificing speed for fuel economy still lets you get to the next station, just a bit later. A refeed is that gas station for your body.
Secondly, eating some more carbohydrates on a refeed day could well assist in your diet throughout the week. Topping your glycogen levels back up will provide some energy for training. And there's evidence of carbohydrate overfeeding actually increasing blood plasma levels of leptin (the satiety hormone staving off hunger) whereas fat does not (see note ).
Personally I can attest for the usefulness of having at least one refeed per week when you are at already fairly lean levels of body composition. Better workouts, better sleep, an easier time to keep weight loss going and a chance to indulge in some food I would normally not eat over the week. It never "messes up" my diet. The past weekend I was abroad with friends for 4 days and basically ate what I wanted. I weighed 3 kg more after that weekend. This weight is almost fully gone again now, 3 days later. Most of it is water retention from having eaten a lot more carbs than usual.
Now, does your friend have a refeed day? I'd say no, it's more like an actual "cheat" day. Eating up to 10000 kcal of ice cream is just excessive. A proper refeed should be part of your nutrition plan and fit into it, it's not a free-for-all buffet. Even if it's not particularly planned (for example, you're going out to eat and don't know what you'll be getting) some restraint should be practiced.
Is this safe?
"Safe" is pretty relative. I wouldn't say it's wise. I wouldn't say it's disastrous either. Binging on some ice cream seems like a better choice than smoking a pack of cigarettes or downing a six-pack of heavy ale in a day. I strongly doubt that one day per week of eating like that is going to be a death sentence, considering how many people eat crap throughout the week and still manage to be relatively okay.
My advice for your friend is to start thinking of it as a refeed rather than cheat day, keep things in check and maybe use the day to eat a variety of satisfying foods rather than emptying a gallon of ice cream.
Also, search on YouTube for the sketch "Butterfield diet". It's comedy, but it's actually just too perfect for this question not to mention it.
Note : In reality things are a lot more complex than a simple sum. People like to repeat "calories in, calories out" and quote the first law of thermodynamics (conservation of energy) in this context. The reason for this is that a lot of folks simply don't want to believe that their biology isn't highly unusual or they don't have some rare disease that makes them resistant to utilizing body fat, so things have to be hammered into their heads in the simplest, most straightforward way. Obviously the first law of thermodynamics will always hold, it's just that the many ways energy is added and subtracted from the system are misunderstood or underestimated — human beings aren't bomb calorimeters and as you'll have noticed we don't excrete ashes. On top of that, there is a limit to the accuracy of the caloric contents food is labeled with. But for all practical purposes in everyday life, thinking of things in the "calories in, calories out" way and just using data on labels and from sources such as the USDA is sufficient.
Note : The "starvation response" discussion has been done elsewhere. See here: https://fitness.stackexchange.com/a/29886/19986
Note : http://link.springer.com/article/10.1007%2Fs001250050686 ("Carbohydrate intake and short-term regulation of leptin in humans", Diabetologia, February 1997, Volume 40, Issue 3, pp 348-351);
http://www.ncbi.nlm.nih.gov/pubmed/11126336 ("Effects of short-term carbohydrate or fat overfeeding on energy expenditure and plasma leptin concentrations in healthy female subjects", Int J Obes Relat Metab Disord. 2000 Nov;24(11):1413-8)