Time machine backup to an external hard drive

Check with the user guide for this gateway to see if it supports USB drives. Even if it does, it does not necessarily mean that it also support Time Machine backups.

Typically, most gateways with USB ports, use them for diagnostics only. Only a few support both USB drives, and fewer still support Time Machine.

There are only a very few routers that will support Time Machine backups to an external drive connected to the USB port on the router.

Among the brands of which I am aware, only Synology and Asus come to mind, but there may be others.

I would be surprised if an Ethernet to USB adapter would work, but it would be worth a try if you have the adapter or can buy an adapter with return privileges.

You will likely need to format the hard drive in Mac OS Extended (Journaled), also known as HFS+

If you mean, using an USB-to-Ethernet adapter, to connect this drive to your gateway's Ethernet port, then sorry, but the answer is still no.

However, if you use a dedicated Network Attached Storage Device (NAS), you can connect one of these to your gateway's Ethernet port, and then, use the NAS for Time Machine backups. Note that not all NAS devices support Time Machine. FWIW, I currently have one from Synology that does.

Also, note that a NAS is a more pricey option as it is basically a dedicated computer that you can place one or more drives into. However, beyond Time Machine, it can be used for many other services as well.