Can normal PC headsets replace 'aviation' headsets?
Can you physically make it work? Yes. But you have to have an adaptor that fixes several compatibility issues. 

Headphone impedance

Impedance is usually referred to in high and low range. Military headphones are low impedance, 19$\Omega$. Civil aviation uses 300$\Omega$, high impedance headphones. Impedance matching is not vital, but can cause some problems with sound quality. Using a low-Z headset with a high-Z output can cause distortion. A low-Z headset will also be more sensitive, so you would have to turn the volume down. You could run the risk of damaging the headset or your hearing if you have the volume up too far. The headphones you listed in the question are 60$\Omega$, which is somewhat of an in-between. There's no real standard, but low-Z are usually less than 32$\Omega$ And high-Z is usually above 100$\Omega$. 

In the recording studio we considered high-Z headsets to be "universal" meaning that you could plug them in anywhere and they would work, but might be limited in volume. I used a set of low-z phones, since I could plug them into a line output and actually hear it. I was always careful to make sure volumes were turned down before I plugged them into the com boxes. They were pretty sensitive.

Microphone impedance

This will make a bigger difference than the headphones. Military systems use 5$\Omega$, low impedance mics. Civilian systems use high-Z, 80-300$\Omega$ amplified electret Mics. The headset you listed in the question lists 2.2k$\Omega$ which would be work, depending on how much amplification it has. Aircraft coms use a vox circuit to cut out noise that opens when you speak. Depending on whether there is a way to adjust the mic input the level may be too much or too little for the vox system. Too little and it won't open when you speak, too low and it will stay open and you will hear all the aircraft noise. 

Phantom power

Electret mics require a power source to operate. They can either contain a battery or run off phantom power, which is a voltage supplied by the device it's plugged into. Some aircraft intercoms provide phantom power. I'm unable to find any standard voltage, but it appears the David Clark aviation headsets run on 8-16v. The standard voltage from a PC audio card is 5v. The headset you listed in the question shows that it requires 2v. I presume that, since it's made for PC use it's intended for 5v, but will operate as low as 2v. There's no way to know if 8-16v would overload it, so you would have to either adapt the phantom power or run it on a battery. 

Connectors

Aviation mics have different physical plugs. 

Most commonly the headset itself uses a 1/4" phono plug, wired the same way as older audio headsets with 1/4" plugs.



The microphones in aviation headsets are seperate and use a somewhat aviation-specific .206" TRS plug. 



Helicopters and military aircraft use a completely different .281" TRRS plug for both headset and mic.



Airbus uses a 5-pin XLR connector for both mic and headset. 



Judging from the fact that people have schematics posted on the web on how to adapt aviation headphones for non-aviation use, but not the other way around I would speculate that even the best non-av sets don't meet the challenges of aviation. Some of the physical design characteristics, such as the pressure exerted on the head pads, etc. are probably quite different from the non-av sets. 

If you look at prices, aviation headsets run well more than twice the price of equivalent quality non-av sets. I'm sure some of that extra price is in design characteristics of the av sets. But, I also suspect that manufacturers will take the exact same sets, change a couple of resistors, put different connectors on them, add the word "aviation" to the box then add $500 to the price. 



Images source

Aviation data found at Acousticon website