If the starter doesn't even click, yet the lights are bright then the batt isn't the problem and I highly doubt the starter died while doing your install.
I had the same issue last week with a newe Chevy Impala. Apparently the car has the LAN set up where the radio is tied into the computer.
Car started first time and we did some settings on the HU/amp. He shut it off then it would do what you described, turn the key and nothing, not even trying to start. He realized there was no door chime and the HU stayed on when he opened the door. If he powered off the HU the car would start.
I didn't do the initial install, but when I was hooking up the comp/sub amps there were 2 wires that appeared to be the REM wires, so I hooked them back up. Apparently the one wire was not supposed to be connected as once I disconnected it the car worked fine.
I believe I had a similar issue doing the HU in my GF's Cobalt and had to tie into a certain place for the REM wire to avoid the same issues...
Cliff's Notes: make sure the HU is powerd off and try to start the car