CentOS

Haven't seen this one before.

nmcli con show public | grep ipv4.gateway

ipv4.gateway: --

nmcli con mod public ipv4.gateway "192.168.1.254"

(no errors, etc., returns to shell prompt)

nmcli con show public | grep ipv4.gateway

ipv4.gateway: --

None of the changes I attempt to make to the public interface stick. The ipv4.method is currently set to manual.

Any ideas why or how to fix it?

Usually one does not need to look at the files, but in this case ... what is in the file now?
That would be /etc/sysconfig/network-scripts/ifcfg-public, I presume.

You did run the commands as root?

Also the ipv4.gateway needs to be within the subnet allocated to the interface or it will not be able to reach it.