Is it code to run a metal water supply line straight through a wall to a refrigerator? It is being run form the cold water outlet of our water heater.

How far is your fridge from the water heater?

Am not sure where you leave, the only thing I found the following statement.

Just to be sure... is your copper of the smooth walled type, or is it corrugated? (If corrugated, you're limited to 10', per California Plumbing Code 604.12.)

I have used supply lines see the link below, and copper tubing.

Everbilt 1/4 in. x 25 ft. Poly Ice Maker Kit-7252-25-14-KIT-EB - The Home Depot

" cold water outlet??". Cold goes in. All house water comes from the same supply line. Or am I missing something?

Code will only require a shutoff for the water line... that's it. Sounds like you have a copper water line going from the water heater cold supply through a wall and connecting to the 'fridge. All sounds acceptable from here!

Mark

In my area of operation, Inspectors always ask for copper tubing - as opposed to plastic tubing - if the tubing is to be exposed and / or ran concealed through the walls & ceilings between supply and the appliance. Also, it has to be protected by nail-plates, treating it just like any other copper water pipe

Hope that helps

Milo