Current and Voltage in CMOS Logic Gate
Logic levels are voltages with respect to a common reference. We usually call that reference ground, despite the fact it confuses almost every newcomer. As such, whatever is receiving this output also needs to be connected or referenced to the same ground or it wont work. 

Shorting the output to Vdd with the output high wont make any difference other than being able to supply more current to whatever the output is connected to. 

If you try to pull the output low when it is shorted to Vdd however, one or both of the bottom transistors will blow.

Similarly, if you short the output to ground. With both inputs high nothing much happens. With any inputs low, one or both of the top transistor will release the "magic smoke".