Not sure of the setup here.
External drive is connected to.. which, the Acer, or the Gateway?
TV is connected to.. which, the Acer or the Gateway?
I assume that the TV is connected to the Gateway, which is the Desktop. So that means the Acer is the laptop, and it is the Acer laptop that has the external drive connected to it.
I'm thinking that, once you shut down the laptop, the power to the external dies and the disk stops spinning. Is there a power switch on the external drive case itself? If so, try "disconnecting hardware" on the Acer, then when Windows says it is safe to remove the hardware, turn the external drive off. Then shut down the Acer.
By the way, do you have backup copies of all those files on the external? If not I would wonder what you would do if the external drive fails.
So I recenlty bought a new computer:
old was 5 year old gateway with xp - ran good just old