Negative, ghost rider. You cannot simply switch a hard drive to a different computer and expect they OS to run the same. However, and depending on whether or not you have an OEM or retail key, you can use software such as ProduKey to get your Windows key off of the old HDD, install a downloaded Windows 7 ISO onto a bootable drive, and then use the aforementioned key to activate Windows. This should work for you. The reason you cannot switch hard drives is because each system uses unique drivers specifically tailored to the hardware.
1. What license is the existing Windows 7? OEM or Retail? If OEM, then it is licensed to the original motherboard. At best, you'd have to call Microsoft and maybe, possibly, they would authorize that license on a new PC. But they are under no obligation to do so.
2. New system = new install. That Windows install that exists on the old drive probably wouldn't even start on the new system.
Sell the old PC, and use the funds to buy the new Windows license for the new PC.
The OS is just as much a part of the PC as is the RAM/GPU/CPU/whatever.