How that light works is on the web so I'll start with no and then suggest you look for this PC's options about Wake On LAN. (AKA WOL)
From WOL to work the Ethernet link has to be "UP" and the link light would be lit.
This has yet to cause an issue since I deployed Ethernet in the 1980's.
I have worn out several DSL modems of various brands over the years but one thing that they had in common is that the ethernet light only came on when the modem (currently a Motorola 3360) was communicating with the computer. Lately, I noticed that the ethernet light has started to come on before I turn the computer (a PC) on. I can make it go out by unplugging the ethernet cable or by unplugging the computer's power cord. Is this a sign of a problem? If so, could this problem also be causing some of the multitude of problems I have with the PC? If so, how do I fix it?