Replacing a new capacity battery and re-calibration I am using redmi 2 for about 3 year and it's battery is dead. It was using 2200 mAH battery. 

So I went to local vendor and bought a copy battery with the rating of 2000 mAH. 

Now the new battery has less capacity of 200 mAH. But my smartphone is calibrated for 2200 mAH battery. So my smartphone will charge the new battery 200 mAH more. 
This will deteoriorate the newer battery.  

So I need way to calibrate the smartphone manually to 2000 mAH so to prevent overcharge of the newer battery. Is there any way to do this with using application and without rooting.
You can't recalibrate, even if your device is rooted since the battery capacity files are coded in the kernel (see Where are the battery capacity files located?)
Since you don't want to install another app , the only preferred option is to make sure that you are alert and switch off the battery when it reaches 90% (difference of 200 mAh being 10% roughly). Reason for saying preferred option is that while the battery should stop charging when it's fully charged, it appears (to me at least) that the hard coded battery capacity files have a role to play on some cases. This is to avoid possible overheating of battery
This can be inconvenient, so you would need an alarm to alert you that it has reached 90% level. You can do this by automation or by using an app like this, which allows you to set an alarm for custom battery charging level ( I haven't tried that app out, picked randomly from many such apps on Play Store)
You may like to read this Ideal charging / discharging percentage for maximum battery life? and set the alarm lower around 80%, if you are keen on battery longevity. This is a theory that is being debated , so it's your choice