Is eating fruits separately healthier?
This answer isn't by far exhausting, but from what I've been able to find…
Let's try to break down the individual claims and compare them with actual studies. Because it is hard to find studies specifically on fruits, I used studies that focused on dietary fiber, sugar and/or vitamins – things fruits are rich in. 
Does mixing fruits with other food slow down gastric emptying?
If you eat fruit close to a meal […] it's held in the stomach too long
I couldn't find any comparison of digest times of ‘fruit only’ vs. ‘fruit with other meal’. However, I found a study (Yu K., 2014) which compares digestion of meals with added dietary fiber to those without added fiber. Fiber was served as a drink with 7.5 g of dietary soluble fiber in it, which is slightly more than the amount of fiber found in 1½ medium apple (USDA). The study found an increase in the time it takes before the stomach to empties half its contents from 72 minutes to 86 minutes.
So while I can't verify the claim that “fruit alone is digested faster than fruit and a meal“ (which is probably also true since larger meals generally take longer time to digest), it has been demonstrated that “meal alone is digested faster than with fruit”.
Is slower gastric emptying a bad thing?
According to Nutrition by Erin, (I'm going to paraphrase it a little) the gastric emptying rate varies between people and depends on many factors. It can be too rapid (symptoms include weakness, diarrhea, and lightheadedness after a meal) or too slow (nausea, vomiting, abdominal pain, and fullness). If one's emptying rate is too slow, they should logically avoid things that slow it down even further, if one's rate is too fast, they can change their diet to help slow it down – eg. they can eat high-fiber foods. Otherwise, some variation in emptying rate is natural.
If you eat fruit close to a meal [it] will rot and ferment in the gut.
A meta-study by R. H. Hunt, 2015 states: “the stomach, together with the oesophagus and duodenum, is the least colonised region of the GI tract”, effectively only H. pylori can live in the acidic environment.
While there may, or may not be a link between slower gastric emptying and H. pylori infection, it is reasoned that the infection causes slower gastric emptying, not the other way round.
In summary, it seems improbable that prolonged digestion in the stomach of a healthy individual, where no bacterial rotting and fermentation can occur, would cause rotting or fermentation in the intestine.
Does empty stomach process the fruits better?
When fruits are eaten alone, your stomach can more easily process all of the nutrients, fibre and the simple sugars contained in the fruit.
This is obviously hard to tell, because monitoring the nutrients of a food as it's being digested isn't a very straightforward task that you could do with ultrasound or X-ray. However there doesn't seem to be much reason for the claim to be true.
Vitamins: Water-soluble vitamins don't need any acids to disolve, fat-soluble vitamins do need special bile acids, but those are found in the intestine, not stomach. 
Fiber: Fruits have large amount of water-soluble fiber, higher than vegetable , so its digestion should be no problem. Not only that, soluble fiber “softens stool and helps slow the movement of the digested material through the digestive tract. This increases nutrient absorption and provides a feeling of fullness for a longer period of time, therefore reducing overall calorie intake. Soluble fiber also binds to bile acids in the intestines and promotes their excretion.” Then there is insoluble fiber which can't be digested at all.
Sugars: Since the other food probably contains more complex carbohydrates, I don't see why there should be any problem with the digestion of the simple sugars found in the fruit. Furthermore, a recent study suggests that fructose, which had been thought to be processed in liver, is in fact mostly processed in the intestine. Only when the amount of fructose is too large, it is sent to the liver, which might be harmful. Therefore the autor suggests that sugar-rich food should be consumed after a meal, not on an empty stomach. However the study was done on mice and the real consequences aren't exactly well studied, so this might be a bit of a stretch.
While it seems to be true that mixing fruits with other meal prolongs the time that the food spends in one's stomach, there doesn't seem to be anything bad about it.