cmos

i would like to know what will happen when the output of a basic CMOS inverter is connected to the ground

Depends on the inverter. 74HC04? 74HCT04? 4049?
Nothing will happen if the input is tied high.

I think you already know the answer. With the input low the output will try to source current into the grounded output. The chip may get quite warm and fail.

It depends on the output current from the type of Cmos IC.
A CD4069 inverter has a typical current into a short of only 4ma when its supply is 5V. Its output transistors can dissipate a max power of 100mW so the 20mW in this case is safe. When the supply is 10V then the typical current into a short is 20mA so the resulting 200mW power dissipation might fry its output transistor.

74HCxx high-speed Cmos ICs have a max allowed output of 25mA but will try to feed 60mA into a short then will fry.