No.

HDMI does have a feature called Ethernet over HDMI that in theory could allow that.

Thing is though it’s literally never been implemented in anything. It died because cheap WiFi became common.

For it to work you’d need both the TV and Chromecast and HDMI cable all to support it. It’s not uncommon on cables and a surprising amount of them include it in features list (probably to trick low info people).

But I believe that’s a hardware design thing so not something even a software update could enable. It costs extra money and they’re already paying for a WiFi chip so why bother?